# questions

tags: ['Physics', 'Gravitation', 'BITSAT', 'BITSAT 2023 (Memory Based Paper 2)']
source_db: MarksBatch2_P1.db
title:
if-v-is-the-gravitational-potential-due-to-sphere-of-uniform-density-on-its-surface-then-its-value-at-the-center-of-sphere-will-be

slug:
if-v-is-the-gravitational-potential-due-to-sphere-of-uniform-density-on-its-surface-then-its-value-at-the-center-of-sphere-will-be-77179

question:
<div class="question">If $\mathrm{V}$ is the gravitational potential due to sphere of uniform density on it's surface, then it's value at the center of sphere will be:</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$\frac{3 V}{2}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><br/>V</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\frac{4}{3} \mathrm{~V}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\frac{\mathrm{V}}{2}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\frac{3 V}{2}$</span> </div>

answer:
<div class="solution">Potential due to sphere<br/>$\mathrm{V}=\frac{\mathrm{GM}}{2 \mathrm{R}^3}\left(3 \mathrm{R}^2-\mathrm{r}^2\right)$<br/>At surface $r=R \Rightarrow V=\left(\frac{G M}{R}\right)$<br/>At centre, $r=0$<br/>$\therefore \mathrm{V}_0=\frac{3 \mathrm{GM}}{2 \mathrm{R}}=\left(\frac{3 \mathrm{~V}}{2}\right)$</div>


title: if-veca-times-vecb2veca-cdot-vecb2676-and-vecb2-then-veca-is-equal-to
slug: if-veca-times-vecb2veca-cdot-vecb2676-and-vecb2-then-veca-is-equal-to-12449
tags: ['Mathematics', 'Vector Algebra', 'BITSAT', 'BITSAT 2010']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">13</span> </div>
source_db: MarksBatch2_P1.db
question:
<div class="question">If \((\vec{a} \times \vec{b})^2+(\vec{a} \cdot \vec{b})^2=676\) and \(|\vec{b}|=2\) then \(|\vec{a}|\) is equal to</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">13</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">26</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">39</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">None of these</span> </li> </ul>

answer:
<div class="solution">\(\begin{aligned} &amp;(\vec{a} \times \vec{b})^2+(\vec{a} \cdot \vec{b})^2=676 \\ &amp;(|\vec{a}| \cdot|\vec{b}| \sin \theta \hat{n})^2+(|\vec{a}| \cdot|\vec{b}| \cos \theta)^2=676 \\ &amp; \Rightarrow a^2 b^2 \sin ^2 \theta+a^2 b^2 \cos ^2 \theta=676\left[(\hat{n})^2=1\right] \\ &amp; a^2 b^2\left(\sin ^2 \theta+\cos ^2 \theta\right)=676 \Rightarrow a^2=\frac{676}{b^2}=\frac{676}{4} \\ &amp;|\vec{a}|=\sqrt{\frac{676}{4}} \Rightarrow|\vec{a}|=\frac{26}{2} \Rightarrow|\vec{a}|=13\end{aligned}\)</div>


title: if-vectors-2-i-j-k-i-2-j-3-k-and-3-i-aj-5-k-are-coplanar-then-the-value-of-a-is
slug: if-vectors-2-i-j-k-i-2-j-3-k-and-3-i-aj-5-k-are-coplanar-then-the-value-of-a-is-12649
tags: ['Mathematics', 'Vector Algebra', 'BITSAT', 'BITSAT 2013']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">-4</span> </div>
source_db: MarksBatch2_P1.db
question:
<div class="question">If vectors $2 \mathrm{i}-\mathrm{j}+\mathrm{k}, \mathrm{i}+2 \mathrm{j}-3 \mathrm{k}$ and $3 \mathrm{i}+\mathrm{aj}+5 \mathrm{k}$ are
<br/> <br/>coplanar, then the value of a is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">2</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">-2</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">-1</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">-4</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

answer:
<div class="solution">If given vectors are coplanar, then there exists two scalar quantities $\mathrm{x}$ and $\mathrm{y}$ such that $2 \hat{i}-\hat{j}+\hat{k}=x(\hat{i}+2 \hat{j}-3 \hat{k})+y(3 \hat{i}+a \hat{j}+5 \hat{k})$
<br/> <br/>Comparing coefficient of $\hat{i}, \hat{j}$ and $\hat{k}$ on both sides of (1) we get $x+3 y=2,2 x+a y=-1,-3 x+5 y=1 \ldots(2)$
<br/> <br/>Solving first and third equations, we get $\mathrm{x}=1 / 2, \mathrm{y}=1 / 2$
<br/> <br/>Since the vectors are coplanar, therefore these values of $x$ and $y$ will satisfy the equation $2 x+a y=-1$
<br/> <br/>$\therefore 2(1 / 2)+a(1 / 2)=-1 \Rightarrow a=-4$</div>


tags: ['Mathematics', 'Parabola', 'BITSAT', 'BITSAT 2023 (Memory Based Paper 2)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value"><br/>8</span> </div>
source_db: MarksBatch2_P1.db
title:
if-vertex-of-a-parabola-is-2-1-and-the-equation-of-its-directrix-is-4-x-3-y-21-then-the-length-of-its-latus-rectum-is

slug:
if-vertex-of-a-parabola-is-2-1-and-the-equation-of-its-directrix-is-4-x-3-y-21-then-the-length-of-its-latus-rectum-is-90159

question:
<div class="question">If vertex of a parabola is $(2,-1)$ and the equation of its directrix is $4 x-3 y=21$, then the length of its latus rectum is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><br/>2</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data"><br/>8</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data"><br/>12</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><br/>16</span> </li> </ul>

answer:
<div class="solution"><br/><img src="https://cdn-question-pool.getmarks.app/pyq/bitsat/Jc4A2EcuWpvDYNl2WTCINIf_f2JCSHl9TOE5Qf0ooo0.original.fullsize.png"/><br/><br/>We are given that directrix of the parabola is $4 x-3 y=21$ and vertex of the parabola is $(2,-1)$<br/>Now $a=\frac{|8+3-21|}{5}=\frac{10}{5}=2$<br/>$\therefore \quad$ latus rectum of the parabola $=4 a=8$</div>


title: if-w-2-then-the-set-of-points-z-w-w-1-is-contained-in-or-equal-to-the-set-of-points-z-satisfying
tags: ['Mathematics', 'Complex Number', 'BITSAT', 'BITSAT 2022']
source_db: MarksBatch2_P1.db
slug:
if-w-2-then-the-set-of-points-z-w-w-1-is-contained-in-or-equal-to-the-set-of-points-z-satisfying-40401

question:
<div class="question">If $|w|=2$, then the set of points $z=w-\frac{1}{w}$ is contained in or equal to the set of points $z$ satisfying</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"> <br/>$\operatorname{Im}(z)=0$
<br/></span> </li><li class=""> <span class="option-label">B</span> <span class="option-data"> <br/>$|\operatorname{Im}(z)| \leq 1$
<br/></span> </li><li class=""> <span class="option-label">C</span> <span class="option-data"> <br/>$|\operatorname{Re}(z)| \leq 2$
<br/></span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data"> <br/>$|z| \leq 3$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"> <br/>$|z| \leq 3$</span> </div>

answer:
<div class="solution">We have, $|w|=2 \Rightarrow w=2(\cos \theta+i \sin \theta)$
<br/>$\therefore \quad z=w-\frac{1}{w}$
<br/>$=2(\cos \theta+i \sin \theta)-\frac{1}{2}(\cos \theta-i \sin \theta)$
<br/>$\Rightarrow \quad z=\frac{3}{2} \cos \theta+\frac{5}{2} i \sin \theta$
<br/>$\Rightarrow \quad|z|=\sqrt{\frac{9}{4}+\frac{25}{4}}=\frac{\sqrt{17}}{2} &lt; 3$
<br/>Hence, option (d) is correct.</div>


title: if-x-0-2-then-the-value-of-cos-1-2-7-1-cos-2-x-sin-2-x-48-cos-2-x-sin-x-is-equal-to
slug: if-x-0-2-then-the-value-of-cos-1-2-7-1-cos-2-x-sin-2-x-48-cos-2-x-sin-x-is-equal-to-36817
tags: ['Mathematics', 'Inverse Trigonometric Functions', 'BITSAT', 'BITSAT 2022']
source_db: MarksBatch2_P1.db
question:
<div class="question">If $x \in\left(0, \frac{\pi}{2}\right)$ then the value of $\cos ^{-1}\left(\frac{7}{2}(1+\cos 2 x)+\sqrt{\left(\sin ^2 x-48 \cos ^2 x\right) \sin x}\right)$ is equal to</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$x-\cos ^{-1}(7 \cos x)$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$x+\sin ^{-1}(7 \cos x)$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$x+\cos ^{-1}(6 \cos x)$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$x+\cos ^{-1}(7 \cos x)$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$x-\cos ^{-1}(7 \cos x)$</span> </div>

answer:
<div class="solution">$y=\cos ^{-1}\left(\frac{7}{2}(1+\cos 2 x)+\sqrt{\left(\sin ^2 x-48 \cos ^2 x\right)} \sin x\right)$
<br/>$=\cos ^{-1}\left((7 \cos x)(\cos x)+\sqrt{1-49 \cos ^2 x} \sqrt{1-\cos ^2 x}\right)$
<br/>$=\cos ^{-1}(\cos x)-\cos ^{-1}(7 \cos x) \quad[\because \cos x &lt; 7 \cos x]$
<br/>$=x-\cos ^{-1}(7 \cos x)$</div>


title: if-x-1-t-2-1-t-2-and-y-1-t-2-2-t-then-d-x-d-y-is-equal-to
slug: if-x-1-t-2-1-t-2-and-y-1-t-2-2-t-then-d-x-d-y-is-equal-to-45688
tags: ['Mathematics', 'Differentiation', 'BITSAT', 'BITSAT 2021']
source_db: MarksBatch2_P1.db
question:
<div class="question">If $x=\frac{1-t^{2}}{1+t^{2}}$ and $y=\frac{2 t}{1+t^{2}}$, then $\frac{d y}{d x}$ is equal to :</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$-\frac{y}{x}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\frac{y}{x}$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$-\frac{x}{y}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\frac{x}{y}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$-\frac{x}{y}$</span> </div>

answer:
<div class="solution">Let $x=\frac{1-t^{2}}{1+t^{2}}$ and $y=\frac{2 t}{1+t^{2}}$
<br/> <br/>Put $t=\tan \theta$, we get
<br/> <br/>$x=\frac{1-\tan ^{2} \theta}{1+\tan ^{2} \theta}$ and $y=\frac{2 \tan \theta}{1+\tan ^{2} \theta}$
<br/> <br/>$\Rightarrow x=\cos 2 \theta$ and $y=\sin 2 \theta$
<br/> <br/>$\therefore \frac{\mathrm{dx}}{\mathrm{d} \theta}=-2 \sin 2 \theta$ and $\frac{\mathrm{dy}}{\mathrm{d} \theta}=2 \cos 2 \theta$
<br/> <br/>Now, $\frac{\mathrm{dy}}{\mathrm{dx}}=\frac{\mathrm{dy}}{\mathrm{d} \theta} \times \frac{\mathrm{d} \theta}{\mathrm{dx}}=-\frac{\cos 2 \theta}{\sin 2 \theta}=-\frac{x}{y}$</div>


title: if-x-2-2-then-the-value-of-x-4-3-x-3-2-x-2-11-x-6-is
slug: if-x-2-2-then-the-value-of-x-4-3-x-3-2-x-2-11-x-6-is-66060
tags: ['Mathematics', 'Complex Number', 'BITSAT', 'BITSAT 2012']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">1</span> </div>
source_db: MarksBatch2_P1.db
question:
<div class="question">If $x=\omega-\omega^{2}-2,$ then the value of $x^{4}+3 x^{3}+2 x^{2}-11 x-6$ is</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">1</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">-1</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">2</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">None of these</span> </li> </ul>

answer:
<div class="solution">We have, $x=\omega-\omega^{2}-2$ or $x+2=\omega-\omega^{2}$
<br/>Squaring, $x^{2}+4 x+4=\omega^{2}+\omega^{4}-2 \omega^{3}$
<br/>$=\omega^{2}+\omega^{3} \omega \cdot-2 \omega^{3}=\omega^{2}+\omega-2 \quad\left[\omega^{3}=1\right]$
<br/>$=-1-2=-3 \Rightarrow$
<br/>$x^{2}+4 x+7=0$
<br/>Dividing $x^{4}+3 x^{3}+2 x^{2}-11 x-6$ by $x^{2}+4 x+7$
<br/>we get $x^{4}+3 x^{3}+2 x^{2}-11 x-6=\left(x^{2}+4 x+7\right)\left(x^{2}-x-1\right)+1$
<br/>$=(0)\left(\mathrm{x}^{2}-\mathrm{x}-1\right)+1=0+1=1$</div>


title: if-x-2-sin-cos-0-cos-sin-1-sin-cos-2-then-the-value-of-4-x-2-x-sin-3-2-5-is
slug: if-x-2-sin-cos-0-cos-sin-1-sin-cos-2-then-the-value-of-4-x-2-x-sin-3-2-5-is-93246
tags: ['Mathematics', 'Determinants', 'BITSAT', 'BITSAT 2018']
source_db: MarksBatch2_P1.db
question:
<div class="question">If <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>x</mi><mn>2</mn></msup><mo>=</mo><mfenced close="|" open="|"><mtable><mtr><mtd><mi>sin</mi><mi>θ</mi></mtd><mtd><mi>cos</mi><mi>θ</mi></mtd><mtd><mn>0</mn></mtd></mtr><mtr><mtd><mo>-</mo><mi>cos</mi><mi>θ</mi></mtd><mtd><mi>sin</mi><mi>θ</mi></mtd><mtd><mn>1</mn></mtd></mtr><mtr><mtd><mi>sin</mi><mi>θ</mi></mtd><mtd><mi>cos</mi><mi>θ</mi></mtd><mtd><mn>2</mn></mtd></mtr></mtable></mfenced></math> then the value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>4</mn><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mi>x</mi><mi>sin</mi><mfrac><mrow><mn>3</mn><mi>π</mi></mrow><mn>2</mn></mfrac><mo>+</mo><mn>5</mn></math> is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>13</mn><mo>-</mo><msqrt><mn>2</mn></msqrt></math></span> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>13</mn><mo>+</mo><msqrt><mn>2</mn></msqrt></math></span> </li><li class=""> <span class="option-label">C</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><msqrt><mn>2</mn></msqrt><mo>-</mo><mn>13</mn></math></span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">Both (A) and (B)</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">Both (A) and (B)</span> </div>

answer:
<div class="solution"><p>Given, <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>x</mi><mn>2</mn></msup><mo>=</mo><mfenced close="|" open="|"><mtable><mtr><mtd><mi>sin</mi><mi>θ</mi></mtd><mtd><mi>cos</mi><mi>θ</mi></mtd><mtd><mn>0</mn></mtd></mtr><mtr><mtd><mo>-</mo><mi>cos</mi><mi>θ</mi></mtd><mtd><mi>sin</mi><mi>θ</mi></mtd><mtd><mn>1</mn></mtd></mtr><mtr><mtd><mi>sin</mi><mi>θ</mi></mtd><mtd><mi>cos</mi><mi>θ</mi></mtd><mtd><mn>2</mn></mtd></mtr></mtable></mfenced></math></p><p>On expanding <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced close="|" open="|"><mtable><mtr><mtd><mi>sin</mi><mi>θ</mi></mtd><mtd><mi>cos</mi><mi>θ</mi></mtd><mtd><mn>0</mn></mtd></mtr><mtr><mtd><mo>-</mo><mi>cos</mi><mi>θ</mi></mtd><mtd><mi>sin</mi><mi>θ</mi></mtd><mtd><mn>1</mn></mtd></mtr><mtr><mtd><mi>sin</mi><mi>θ</mi></mtd><mtd><mi>cos</mi><mi>θ</mi></mtd><mtd><mn>2</mn></mtd></mtr></mtable></mfenced></math> along <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>C</mi><mn>3</mn></msub></math>, we get</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>x</mi><mn>2</mn></msup><mo>=</mo><mn>0</mn><mo>-</mo><mn>1</mn><mfenced close="|" open="|"><mtable><mtr><mtd><mi>sin</mi><mi>θ</mi></mtd><mtd><mi>cos</mi><mi>θ</mi></mtd></mtr><mtr><mtd><mi>sin</mi><mi>θ</mi></mtd><mtd><mi>cos</mi><mi>θ</mi></mtd></mtr></mtable></mfenced><mo>+</mo><mn>2</mn><mfenced close="|" open="|"><mtable><mtr><mtd><mi>sin</mi><mi>θ</mi></mtd><mtd><mi>cos</mi><mi>θ</mi></mtd></mtr><mtr><mtd><mo>-</mo><mi>cos</mi><mi>θ</mi></mtd><mtd><mi>sin</mi><mi>θ</mi></mtd></mtr></mtable></mfenced><mo>=</mo><mo>-</mo><mn>1</mn><mfenced><mrow><mi>sin</mi><mi>θ</mi><mi>cos</mi><mi>θ</mi><mo>-</mo><mi>sin</mi><mi>θ</mi><mi>cos</mi><mi>θ</mi></mrow></mfenced><mo>+</mo><mn>2</mn><mfenced><mrow><msup><mi>sin</mi><mn>2</mn></msup><mi>θ</mi><mo>+</mo><msup><mi>cos</mi><mn>2</mn></msup><mi>θ</mi></mrow></mfenced></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mo>-</mo><mn>1</mn><mo>×</mo><mn>0</mn><mo>+</mo><mn>2</mn><mo>×</mo><mn>1</mn></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><msup><mi>x</mi><mn>2</mn></msup><mo>=</mo><mn>2</mn><mo>⇒</mo><mi>x</mi><mo>=</mo><mo>±</mo><msqrt><mn>2</mn></msqrt></math></p><p>If <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><msqrt><mn>2</mn></msqrt></math>, then</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>4</mn><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mi>x</mi><mi>sin</mi><mfrac><mrow><mn>3</mn><mi>π</mi></mrow><mn>2</mn></mfrac><mo>+</mo><mn>5</mn><mo>=</mo><mn>4</mn><mo>×</mo><msup><mfenced><msqrt><mn>2</mn></msqrt></mfenced><mn>2</mn></msup><mo>-</mo><msqrt><mn>2</mn></msqrt><mo>+</mo><mn>5</mn></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>8</mn><mo>-</mo><msqrt><mn>2</mn></msqrt><mo>+</mo><mn>5</mn><mo>=</mo><mfenced><mrow><mn>13</mn><mo>-</mo><msqrt><mn>2</mn></msqrt></mrow></mfenced></math></p><p>If <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mo>-</mo><msqrt><mn>2</mn></msqrt></math>, then</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>4</mn><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mi>x</mi><mi>sin</mi><mfrac><mrow><mn>3</mn><mi>π</mi></mrow><mn>2</mn></mfrac><mo>+</mo><mn>5</mn><mo>=</mo><mn>4</mn><mo>×</mo><msup><mfenced><mrow><mo>-</mo><msqrt><mn>2</mn></msqrt></mrow></mfenced><mn>2</mn></msup><mo>+</mo><msqrt><mn>2</mn></msqrt><mo>+</mo><mn>5</mn></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>8</mn><mo>+</mo><msqrt><mn>2</mn></msqrt><mo>+</mo><mn>5</mn><mo>=</mo><mfenced><mrow><mn>13</mn><mo>+</mo><msqrt><mn>2</mn></msqrt></mrow></mfenced></math></p></div>


title: if-x-3-x-5-3-x-1-d-x-x-3-5-d-x-x-5-b-d-x-then-the-value-of-b-is
slug: if-x-3-x-5-3-x-1-d-x-x-3-5-d-x-x-5-b-d-x-then-the-value-of-b-is-90477
tags: ['Mathematics', 'Indefinite Integration', 'BITSAT', 'BITSAT 2021']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">8</span> </div>
source_db: MarksBatch2_P1.db
question:
<div class="question">If $\int \frac{3 x+1}{(x-3)(x-5)} d x=\int \frac{-5}{(x-3)} d x+\int \frac{B}{(x-5)} d x$ then the value of $B$ is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">3</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">4</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">6</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">8</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

answer:
<div class="solution">We have,
<br/> <br/>$\begin{array}{l}
<br/> <br/>\frac{3 x+1}{(x-3)(x-5)}=\frac{-5}{x-3}+\frac{\mathrm{B}}{x-5} \\
<br/> <br/>3 x+1=-5(x-5)+\mathrm{B}(x-3) \\
<br/> <br/>\text { Put } x=5 \\
<br/> <br/>3(5)+1=\mathrm{B}(5-3) \\
<br/> <br/>16=2 \mathrm{~B} \text { or } \mathrm{B}=8
<br/> <br/>\end{array}$</div>


title: if-x-5-d-x-x-7-x-6-log-x-9-x-9-1-c-then-a-equal-to
slug: if-x-5-d-x-x-7-x-6-log-x-9-x-9-1-c-then-a-equal-to-49844
tags: ['Mathematics', 'Indefinite Integration', 'BITSAT', 'BITSAT 2017']
source_db: MarksBatch2_P1.db
question:
<div class="question">If <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>∫</mo><mfrac><mrow><mo>(</mo><msqrt><mi>x</mi></msqrt><msup><mo>)</mo><mn>5</mn></msup><mi>d</mi><mi>x</mi></mrow><mrow><mo>(</mo><msqrt><mi>x</mi></msqrt><msup><mo>)</mo><mn>7</mn></msup><mo>+</mo><msup><mi>x</mi><mn>6</mn></msup></mrow></mfrac><mo>=</mo><mi>λ</mi><mi>log</mi><mfenced><mfrac><msup><mi>x</mi><mn>9</mn></msup><mrow><msup><mi>x</mi><mn>9</mn></msup><mo>+</mo><mn>1</mn></mrow></mfrac></mfenced><mo>+</mo><mi>C</mi></math>, then <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>+</mo><mi>λ</mi></math> equal to</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn></math></span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>&gt;</mo><mn>2</mn></math></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>&lt;</mo><mn>2</mn></math></span> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>&gt;</mo><mn>3</mn></math></span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>&gt;</mo><mn>2</mn></math></span> </div>

answer:
<div class="solution"><p>Let <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>I</mi><mo>=</mo><mo>∫</mo><mfrac><mrow><mi>d</mi><mi>x</mi></mrow><mrow><msup><mfenced><msqrt><mi>x</mi></msqrt></mfenced><mn>2</mn></msup><mo>+</mo><msup><mfenced><msqrt><mi>x</mi></msqrt></mfenced><mn>7</mn></msup></mrow></mfrac></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mo>∫</mo><mfrac><mrow><mi>d</mi><mi>x</mi></mrow><mrow><msup><mfenced><msqrt><mi>x</mi></msqrt></mfenced><mn>7</mn></msup><mfenced><mrow><mfrac><mn>1</mn><mrow><mo>(</mo><msqrt><mi>x</mi></msqrt><msup><mo>)</mo><mn>5</mn></msup></mrow></mfrac><mo>+</mo><mn>1</mn></mrow></mfenced></mrow></mfrac></math></p><p>Put <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn><mo>+</mo><mfrac><mn>1</mn><msup><mfenced><msqrt><mi>x</mi></msqrt></mfenced><mn>5</mn></msup></mfrac><mo>=</mo><mi>t</mi></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mo> </mo><mo>-</mo><mfrac><mn>5</mn><mn>2</mn></mfrac><mo>·</mo><mfrac><mn>1</mn><mrow><mo>(</mo><msqrt><mi>x</mi></msqrt><msup><mo>)</mo><mn>7</mn></msup></mrow></mfrac><mi>d</mi><mi>x</mi><mo>=</mo><mi>d</mi><mi>t</mi></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>∴</mo><mo> </mo><mi>I</mi><mo>=</mo><mo>-</mo><mfrac><mn>2</mn><mn>5</mn></mfrac><mo>∫</mo><mfrac><mrow><mi>d</mi><mi>t</mi></mrow><mi>t</mi></mfrac><mo>=</mo><mfrac><mrow><mo>-</mo><mn>2</mn></mrow><mn>5</mn></mfrac><mi>log</mi><mi>t</mi><mo>+</mo><mi>C</mi></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfrac><mrow><mo>-</mo><mn>2</mn></mrow><mn>5</mn></mfrac><mi>log</mi><mfenced close="]" open="["><mrow><mn>1</mn><mo>+</mo><mfrac><mn>1</mn><msup><mfenced><msqrt><mi>x</mi></msqrt></mfenced><mn>5</mn></msup></mfrac></mrow></mfenced><mo>+</mo><mi>C</mi></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mo>-</mo><mfrac><mn>2</mn><mn>5</mn></mfrac><mi>log</mi><mfenced><mfrac><mrow><msup><mi>x</mi><mrow><mn>5</mn><mo>/</mo><mn>2</mn></mrow></msup><mo>+</mo><mn>1</mn></mrow><msup><mi>x</mi><mrow><mn>5</mn><mo>/</mo><mn>2</mn></mrow></msup></mfrac></mfenced><mo>+</mo><mi>C</mi></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mo> </mo><mi>λ</mi><mo>=</mo><mfrac><mrow><mo>-</mo><mn>2</mn></mrow><mn>5</mn></mfrac></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>=</mo><mfrac><mn>5</mn><mn>2</mn></mfrac></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>∴</mo><mo> </mo><mi>a</mi><mo>+</mo><mi>λ</mi><mo>=</mo><mn>2</mn><mo>.</mo><mn>1</mn><mo>&gt;</mo><mn>2</mn></math></p></div>


title: if-x-18-occurs-in-the-rth-term-in-the-expansion-of-x-4-x-3-1-15-then-what-is-the-value-of-r
slug: if-x-18-occurs-in-the-rth-term-in-the-expansion-of-x-4-x-3-1-15-then-what-is-the-value-of-r-57521
tags: ['Mathematics', 'Binomial Theorem', 'BITSAT', 'BITSAT 2023 (Memory Based Paper 2)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value"><br/>7</span> </div>
source_db: MarksBatch2_P1.db
question:
<div class="question">If $x^{18}$ occurs in the rth term in the expansion of $\left(x^4+\frac{1}{x^3}\right)^{15}$, then what is the value of $r$ ?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><br/>3</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><br/>5</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data"><br/>7</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><br/>9</span> </li> </ul>

answer:
<div class="solution">In the expansion of $\left(x^4+\frac{1}{x^3}\right)^{15}$, let $T_r$ is the $r^{\text {th }}$ term<br/>$\begin{aligned}<br/>T_r &amp; ={ }^{15} C_{r-1}\left(x^4\right)^{15-r+1}\left(\frac{1}{x^3}\right)^{r-1} \\<br/>&amp; ={ }^{15} C_{r-1} x^{64-4 r-3 r+3}=15_{C_{r-1}} x^{67-7 r}<br/>\end{aligned}$<br/>$\mathrm{x}^{18}$ occurs in this term<br/>$\Rightarrow 18=67-7 r \Rightarrow 7 r=49 \Rightarrow r=7 .$</div>


title: if-x-a-sin-and-y-b-cos-then-d-x-2-d-2-y-is
slug: if-x-a-sin-and-y-b-cos-then-d-x-2-d-2-y-is-75117
tags: ['Mathematics', 'Differentiation', 'BITSAT', 'BITSAT 2012']
source_db: MarksBatch2_P1.db
question:
<div class="question">If $x=a \sin \theta$ and $y=b \cos \theta,$ then $\frac{d^{2} y}{d x^{2}}$ is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\frac{a}{b^{2}} \sec ^{2} \theta$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$\frac{-b}{a} \sec ^{2} \theta$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\frac{-b}{a^{2}} \sec ^{3} \theta$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\frac{b}{a^{2}} \sec ^{3} \theta$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\frac{-b}{a} \sec ^{2} \theta$</span> </div>

answer:
<div class="solution">(b) Given $x=a \sin \theta$ and $y=b \cos \theta$
<br/>$\Rightarrow \frac{d x}{d \theta}=a \cos \theta$ and $\frac{d y}{d \theta}=-b \sin \theta$
<br/>$\therefore \quad \frac{d y}{d x}=\frac{d y}{d \theta} \times \frac{d \theta}{d x}=-\frac{b}{a} \tan \theta \Rightarrow \frac{d^{2} y}{d x^{2}}=\frac{-b}{a} \sec ^{2} \theta$</div>


title: if-x-is-a-differential-function-then-the-solution-of-the-differential-equation-dy-y-x-x-x-d-x-0-is
tags: ['Mathematics', 'Differential Equations', 'BITSAT', 'BITSAT 2015']
source_db: MarksBatch2_P1.db
slug:
if-x-is-a-differential-function-then-the-solution-of-the-differential-equation-dy-y-x-x-x-d-x-0-is-73310

question:
<div class="question">If $\phi(\mathrm{x})$ is a differential function, then the solution of the differential equation $\mathrm{dy}+\left\{\mathrm{y} \phi^{\prime}(\mathrm{x})-\phi(\mathrm{x})\right.$
<br/> <br/>$\left.\phi^{\prime}(x)\right\} d x=0,$ is</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$y=\{\phi(x)-1\}+C e^{-\phi(x)}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$y \phi(x)=\{\phi(x)\}^{2}+C$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$y e^{\varphi(x)}=\phi(x) e^{\phi(x)}+C$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$y-\phi(x)=\phi(x) e^{-\phi(x)}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$y=\{\phi(x)-1\}+C e^{-\phi(x)}$</span> </div>

answer:
<div class="solution">Given differential equation is
<br/> <br/>$\mathrm{dy}+\left\{y \phi^{\prime}(x)-\phi(x) \phi^{\prime}(x)\right\} d x=0$
<br/> <br/>$\Rightarrow \frac{\mathrm{dy}}{\mathrm{dx}}+\phi^{\prime}(\mathrm{x}) \mathrm{y}=\phi(\mathrm{x}) \phi^{\prime}(\mathrm{x})$
<br/> <br/>which is a linear differential equation with $\quad P=\phi^{\prime}(x), Q=\phi(x) \cdot \phi^{\prime}(x)$ and
<br/> <br/>$I \cdot F=e^{\int \phi^{\prime}(x) d x}=e^{\phi(x)}$
<br/> <br/>$\therefore$ Solution is $y \cdot e^{\phi(x)}=\int \phi(x) \cdot \phi^{\prime}(x) e^{\phi(x)} d x+C$
<br/> <br/>$\Rightarrow y \cdot e^{\phi(x)}=\int \phi(x) \cdot e^{\phi(x)} \phi^{\prime}(x) d x+C$
<br/> <br/>$\Rightarrow y \cdot e^{\phi(x)}=\phi(x) e^{\phi(x)}-\int \phi^{\prime}(x) e^{\phi(x)} d x+C$
<br/> <br/>$\Rightarrow y \cdot e^{\phi(x)}=\phi(x) e^{\phi(x)}-e^{\phi(x)}+C$
<br/> <br/>$\Rightarrow y=\lceil\phi(x)-1]+C e^{-\phi(x)}$</div>


tags: ['Mathematics', 'Sequences and Series', 'BITSAT', 'BITSAT 2012']
source_db: MarksBatch2_P1.db
title:
if-x-is-positive-then-the-sum-to-infinity-of-the-series-1-3-x-1-1-3-x-2-1-3-x-1-3-x-3-1-3-x-2-1-3-x-4-1-3-x-3-ldots-is

slug:
if-x-is-positive-then-the-sum-to-infinity-of-the-series-1-3-x-1-1-3-x-2-1-3-x-1-3-x-3-1-3-x-2-1-3-x-4-1-3-x-3-ldots-is-93239

question:
<div class="question">If $x$ is positive then the sum to infinity of the
<br/>series $\frac{1}{1+3 x}-\frac{1-3 x}{(1+3 x)^{2}}+\frac{(1-3 x)^{2}}{(1+3 x)^{3}}-\frac{(1-3 x)^{3}}{(1+3 x)^{4}}$ \ldots $\infty$ is</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$1 / 2$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\frac{1}{6 x}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\frac{1}{6 x(1+3 x)}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\frac{1}{2(1+3 x)}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$1 / 2$</span> </div>

answer:
<div class="solution">The series is a G.P. with common ratio
<br/>$=\left(\frac{1-3 \mathrm{x}}{1+3 \mathrm{x}}\right)$ and $|\mathrm{r}|=\left|\frac{1-3 \mathrm{x}}{1+3 \mathrm{x}}\right|$ is less than 1 since $\mathrm{x}$ is
<br/>positive $\mathrm{S}_{\infty}=\frac{\mathrm{a}}{1-\mathrm{r}}=\frac{\frac{1}{1+3 \mathrm{x}}}{1-\left\{-\left(\frac{1-3 \mathrm{x}}{1+3 \mathrm{x}}\right)\right\}}=\frac{1}{2}$</div>


title: if-x-log-5-3-log-7-5-log-9-7-then
slug: if-x-log-5-3-log-7-5-log-9-7-then-47130
tags: ['Mathematics', 'Basic of Mathematics', 'BITSAT', 'BITSAT 2019']
source_db: MarksBatch2_P1.db
question:
<div class="question">If <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><msub><mi>log</mi><mn>5</mn></msub><mn>3</mn><mo>+</mo><msub><mi>log</mi><mn>7</mn></msub><mn>5</mn><mo>+</mo><msub><mi>log</mi><mn>9</mn></msub><mn>7</mn></math>, then</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>&gt;</mo><mfrac><mn>3</mn><mn>2</mn></mfrac></math></span> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>&gt;</mo><mfrac><mn>1</mn><mroot><mn>2</mn><mn>3</mn></mroot></mfrac></math></span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>&gt;</mo><mfrac><mn>3</mn><mroot><mn>2</mn><mn>3</mn></mroot></mfrac></math></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>&gt;</mo><mroot><mn>2</mn><mn>3</mn></mroot></math></span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>&gt;</mo><mfrac><mn>3</mn><mroot><mn>2</mn><mn>3</mn></mroot></mfrac></math></span> </div>

answer:
<div class="solution"><blockquote><p>We have, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><msub><mi>log</mi><mn>5</mn></msub><mn>3</mn><mo>+</mo><msub><mi>log</mi><mn>7</mn></msub><mn>5</mn><mo>+</mo><msub><mi>log</mi><mn>9</mn></msub><mn>7</mn></math></p><p>Using <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>AM</mi><mo>≥</mo><mi>GM</mi></math> for three positive real numbers <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mi>a</mi><mo>+</mo><mi>b</mi><mo>+</mo><mi>c</mi></mrow><mn>3</mn></mfrac><mo>≥</mo><msup><mfenced><mrow><mi>a</mi><mi>b</mi><mi>c</mi></mrow></mfenced><mfrac bevelled="true"><mn>1</mn><mn>3</mn></mfrac></msup></math>. Equality holds when <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>=</mo><mi>b</mi><mo>=</mo><mi>c</mi><mo>.</mo></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>∴</mo><mo> </mo><mfrac><mrow><msub><mi>log</mi><mn>5</mn></msub><mn>3</mn><mo>+</mo><msub><mi>log</mi><mn>7</mn></msub><mn>5</mn><mo>+</mo><msub><mi>log</mi><mn>9</mn></msub><mn>7</mn></mrow><mn>3</mn></mfrac><mo>&gt;</mo><msup><mfenced><mrow><msub><mi>log</mi><mn>5</mn></msub><mn>3</mn><mo>·</mo><msub><mi>log</mi><mn>7</mn></msub><mn>5</mn><mo>·</mo><msub><mi>log</mi><mn>9</mn></msub><mn>7</mn></mrow></mfenced><mrow><mn>1</mn><mo>/</mo><mn>3</mn></mrow></msup></math><math xmlns="http://www.w3.org/1998/Math/MathML"></math><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mo> </mo><mi>x</mi><mo>&gt;</mo><mn>3</mn><msup><mfenced><mrow><mfrac><mrow><mi>log</mi><mn>3</mn></mrow><mrow><mi>log</mi><mn>5</mn></mrow></mfrac><mo>×</mo><mfrac><mrow><mi>log</mi><mn>5</mn></mrow><mrow><mi>log</mi><mn>7</mn></mrow></mfrac><mo>×</mo><mfrac><mrow><mi>log</mi><mn>7</mn></mrow><mrow><mi>log</mi><mn>9</mn></mrow></mfrac></mrow></mfenced><mrow><mn>1</mn><mo>/</mo><mn>3</mn></mrow></msup></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mo> </mo><mi>x</mi><mo>&gt;</mo><mn>3</mn><msup><mfenced><mfrac><mn>1</mn><mn>2</mn></mfrac></mfenced><mrow><mn>1</mn><mo>/</mo><mn>3</mn></mrow></msup><mo>⇒</mo><mi>x</mi><mo>&gt;</mo><mfrac><mn>3</mn><mroot><mn>2</mn><mn>3</mn></mroot></mfrac></math></p></blockquote></div>


title: if-x-satisfies-3-x-2-3-x-4-3-x-6-12-then
slug: if-x-satisfies-3-x-2-3-x-4-3-x-6-12-then-24459
question: <div class="question">If x satisfies $|3 x-2|+|3 x-4|+|3 x-6| \geq 12$, then</div>
tags: ['Mathematics', 'Basic of Mathematics', 'BITSAT', 'BITSAT 2013']
source_db: MarksBatch2_P1.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$0 \leq \mathrm{x} \geq \frac{8}{3}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$x \geq \frac{8}{3}$ or $\frac{-4}{3}$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$x \leq 0$ or $x \geq \frac{8}{3}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$x \geq 2$ only</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$x \leq 0$ or $x \geq \frac{8}{3}$</span> </div>

answer:
<div class="solution">Dividing $\mathrm{R}$ at $\frac{2}{3}, \frac{4}{3}$ and $2,$ analyse 4 cases. When $\mathrm{x} \leq \frac{2}{3},$ the inequality becomes $2-3 x+4-3 x+6-3 x \geq 12$
<br/> <br/>implying $-9 x \geq 0 \Rightarrow x \leq 0$
<br/> <br/>when $\mathrm{x} \geq 2$ the ineqality becomes $3 x-2+3 x-4+3 x-6 \geq 12$
<br/> <br/>Implying $9 \mathrm{x} \geq 24 \Rightarrow \mathrm{x} \geq 8 / 3$
<br/> <br/>The inequality in invalid in the other two sections.
<br/> <br/>$\therefore$ either $x \leq 0$ or $x \geq 8 / 3$</div>


title: if-x-sin-3-y-cos-3-sin-cos-and-x-sin-y-cos-then-x-2-y-2
slug: if-x-sin-3-y-cos-3-sin-cos-and-x-sin-y-cos-then-x-2-y-2-81149
tags: ['Mathematics', 'Trigonometric Ratios & Identities', 'BITSAT', 'BITSAT 2013']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">1</span> </div>
source_db: MarksBatch2_P1.db
question:
<div class="question">If $x \sin ^{3} \theta+y \cos ^{3} \theta=\sin \theta \cos \theta$ and
<br/> <br/>$x \sin \theta=y \cos \theta,$ then $x^{2}+y^{2}=$</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">1</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">2</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">0</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">None of these</span> </li> </ul>

answer:
<div class="solution">$x \sin ^{3} \theta+y \cos ^{3} \theta=\sin \theta \cos \theta$
<br/> <br/>and $x \sin \theta=y \cos \theta$
<br/> <br/>Equation (i) may be written as $x \sin \theta \cdot \sin ^{2} \theta+y \cos ^{3} \theta=\sin \theta \cos \theta$
<br/> <br/>$\Rightarrow y \cos \theta \sin ^{2} \theta+y \cos ^{3} \theta=\sin \theta \cos \theta$
<br/> <br/>$\Rightarrow y \cos \theta\left(\sin ^{2} \theta+\cos ^{2} \theta\right)=\sin \theta \cos \theta$
<br/> <br/>$\Rightarrow y \cos \theta=\sin \theta \cos \theta \therefore y=\sin \theta \quad \ldots$
<br/> <br/>Putting the value of y from
<br/> <br/>(iii) in (ii), we get $\mathrm{x} \sin \theta=\sin \theta \cdot \cos \theta \Rightarrow \mathrm{x}=\cos \theta \quad \ldots$ (iv)
<br/> <br/>Squaring (iii) and (iv) and adding, we get $\mathrm{x}^{2}+\mathrm{y}^{2}=\cos ^{2} \theta+\sin ^{2} \theta=1$</div>


tags: ['Physics', 'Oscillations', 'BITSAT', 'BITSAT 2018']
source_db: MarksBatch2_P1.db
title:
if-x-v-and-a-denote-the-displacement-the-velocity-and-the-acceleration-of-a-particle-executing-s-h-m-of-time-period-t-then-which-of-the-following-does

slug:
if-x-v-and-a-denote-the-displacement-the-velocity-and-the-acceleration-of-a-particle-executing-s-h-m-of-time-period-t-then-which-of-the-following-does-39830

question:
<div class="question">If <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>,</mo><mo> </mo><mi>v</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi></math> denote the displacement, the velocity and the acceleration of a particle executing <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>S</mi><mi>H</mi><mi>M</mi></math> of time period <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>T</mi></math>. Then, which of the following does not change with time?</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mi>a</mi><mi>T</mi></mrow><mi>x</mi></mfrac></math></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mi>T</mi><mo>+</mo><mn>2</mn><mi>π</mi><mi>v</mi></math></span> </li><li class=""> <span class="option-label">C</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mi>a</mi><mi>T</mi></mrow><mi>v</mi></mfrac></math></span> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>a</mi><mn>2</mn></msup><msup><mi>T</mi><mn>2</mn></msup><mo>+</mo><mn>4</mn><msup><mi>π</mi><mn>2</mn></msup><msup><mi>v</mi><mn>2</mn></msup></math></span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mi>a</mi><mi>T</mi></mrow><mi>x</mi></mfrac></math></span> </div>

answer:
<div class="solution"><p>We know, instantaneous displacement,</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mi>r</mi><mo> </mo><mi>sin</mi><mi>ω</mi><mi>t</mi></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>∵</mo></math> Instantaneous velocity, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>v</mi><mo>=</mo><mfrac><mrow><mi>d</mi><mi>x</mi></mrow><mrow><mi>d</mi><mi>t</mi></mrow></mfrac><mo>=</mo><mi>r</mi><mi>ω</mi><mo> </mo><mi>cos</mi><mi>ω</mi><mi>t</mi></math></p><p>So. <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mi>a</mi><mi>T</mi></mrow><mi>x</mi></mfrac><mo>=</mo><mfrac><mrow><mo>-</mo><msup><mi>ω</mi><mn>2</mn></msup><mi>x</mi><mo>×</mo><mi>T</mi></mrow><mi>x</mi></mfrac><mo>=</mo><mo>-</mo><msup><mi>ω</mi><mn>2</mn></msup><mi>T</mi><mo>=</mo><mfrac><mrow><mo>-</mo><mn>4</mn><msup><mi>π</mi><mn>2</mn></msup></mrow><msup><mi>T</mi><mn>2</mn></msup></mfrac><mo>×</mo><mi>T</mi></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfrac><mrow><mo>-</mo><mn>4</mn><msup><mi>π</mi><mn>2</mn></msup></mrow><mi>T</mi></mfrac><mo>=</mo></math> constant</p><p>and,</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mo>  </mo><mfrac><mrow><mi>a</mi><mi>T</mi></mrow><mi>V</mi></mfrac><mo>=</mo><mfrac><mrow><mo>-</mo><msup><mi>ω</mi><mn>2</mn></msup><mi>r</mi><mi>sin</mi><mi>ω</mi><mi>t</mi><mo>×</mo><mi>T</mi></mrow><mrow><mi>ω</mi><mi>r</mi><mi>cos</mi><mi>ω</mi><mi>t</mi></mrow></mfrac><mo>=</mo><mo>-</mo><mi>ω</mi><mi>T</mi><mi>tan</mi><mi>ω</mi><mi>T</mi></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mo>-</mo><mfrac><mrow><mn>2</mn><mi>π</mi></mrow><mi>T</mi></mfrac><mo>×</mo><mi>T</mi><mo> </mo><mi>tan</mi><mi>ω</mi><mi>t</mi><mo>=</mo></math>not constant</p><p>Similarly, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mi>T</mi><mo>+</mo><mn>2</mn><mi>π</mi><mi>v</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>a</mi><mn>2</mn></msup><msup><mi>T</mi><mn>2</mn></msup><mo>+</mo><mn>4</mn><msup><mi>π</mi><mn>2</mn></msup><msup><mi>v</mi><mn>2</mn></msup></math> is also not constant,<br/>i.e. both are function of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi></math></p></div>


title: if-x-y-and-z-are-real-numbers-the-x-2-4-y-2-9-z-2-6-yz-3-z-x-2-x-y-is-always
slug: if-x-y-and-z-are-real-numbers-the-x-2-4-y-2-9-z-2-6-yz-3-z-x-2-x-y-is-always-98014
tags: ['Mathematics', 'Basic of Mathematics', 'BITSAT', 'BITSAT 2020']
source_db: MarksBatch2_P1.db
question:
<div class="question">If $x, y$ and $z$ are real numbers, the $x^{2}+4 y^{2}+9 z^{2}-6 y z-3 z x-2 x y$ is always</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">positive</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">non-positive</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">zero</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">non-negative</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">non-negative</span> </div>

answer:
<div class="solution">$x^{2}+4 y^{2}+9 z^{2}-6 y z-3 z x-2 x y$
<br/> <br/>$=x^{2}+(2 y)^{2}+(3 z)^{2}-(2 y)(3 z)-(3 z)(x)-$ $x(2 y) \geq 0 \forall x, y, z$
<br/> <br/>$\left[\because \mathrm{a}^{2}+\mathrm{b}^{2}+\mathrm{c}^{2}-\mathrm{ab}-\mathrm{bc}-\mathrm{ca} \geq 0\right]$</div>


title: if-x-y-x-2-y-2-1-x-y-r-then-is
slug: if-x-y-x-2-y-2-1-x-y-r-then-is-27150
tags: ['Mathematics', 'Sets and Relations', 'BITSAT', 'BITSAT 2015']
source_db: MarksBatch2_P1.db
question:
<div class="question">If $\rho=\left\{(x, y) \mid x^{2}+y^{2}=1 ; x, y \in \mathbb{R}\right\}$. Then, $\rho$ is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">reflexive</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">symmetric</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">transitive</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">anti-symmetric</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">symmetric</span> </div>

answer:
<div class="solution">Obviously, the relation is not reflexive and transitive, but it is symmetric, because $x^{2}+x^{2}=2 x^{2} \neq 1$
<br/> <br/>$$
<br/> <br/>\begin{array}{l}
<br/> <br/>\text { and } x^{2}+y^{2}=1, y^{2}+z^{2}=1 \\
<br/> <br/>\Rightarrow x^{2}+z^{2}=1 \\
<br/> <br/>\text { But } x^{2}+y^{2}=1 \Rightarrow y^{2}+x^{2}=1
<br/> <br/>\end{array}
<br/> <br/>$$</div>


title: if-x-y-y-2-tan-x-y-then-find-d-x-d-y-is
slug: if-x-y-y-2-tan-x-y-then-find-d-x-d-y-is-64259
question: <div class="question">If $x y+y^{2}=\tan x+y,$ then find $\frac{d y}{d x}$ is</div>
tags: ['Mathematics', 'Differentiation', 'BITSAT', 'BITSAT 2020']
source_db: MarksBatch2_P1.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\frac{\sec ^{2} x}{x+2 y}$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$\frac{\sec ^{2} x-y}{(x+2 y-1)}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$(x+2 y-1) \sec ^{2} x$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\sec ^{2} x \cdot y$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\frac{\sec ^{2} x-y}{(x+2 y-1)}$</span> </div>

answer:
<div class="solution">The given relation is $x y+y^{2}=\tan x+y$. Differentiating both sides with respect to $x$, we get
<br/> <br/>$\frac{d}{d x}(x y)+\frac{d}{d x}\left(y^{2}\right)=\frac{d}{d x}(\tan x)+\frac{d y}{d x}$
<br/> <br/>or $\left[y \cdot 1+x \cdot \frac{d y}{d x}\right]+2 y \frac{d y}{d x}=\sec ^{2} x+\frac{d y}{d x}$
<br/> <br/>or $(x+2 y-1) \frac{d y}{d x}=\sec ^{2} x-y$
<br/> <br/>$\therefore \frac{d y}{d x}=\frac{\sec ^{2} x-y}{(x+2 y-1)}$</div>


title: if-x-y-z-r-x-y-z-5-x-2-y-2-z-2-9-then-length-of-interval-in-which-x-lies-is
slug: if-x-y-z-r-x-y-z-5-x-2-y-2-z-2-9-then-length-of-interval-in-which-x-lies-is-74200
tags: ['Mathematics', 'Quadratic Equation', 'BITSAT', 'BITSAT 2019']
source_db: MarksBatch2_P1.db
question:
<div class="question">If <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>,</mo><mi>y</mi><mo>,</mo><mi>z</mi><mo>∈</mo><mi>R</mi><mo>,</mo><mo> </mo><mi>x</mi><mo>+</mo><mi>y</mi><mo>+</mo><mi>z</mi><mo>=</mo><mn>5</mn><mo>,</mo><mo> </mo><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><msup><mi>y</mi><mn>2</mn></msup><mo>+</mo><msup><mi>z</mi><mn>2</mn></msup><mo>=</mo><mn>9</mn></math>, then length of interval in which <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math> lies is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>8</mn><mo>/</mo><mn>3</mn></math></span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>4</mn><mo>/</mo><mn>3</mn></math></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><mo>/</mo><mn>3</mn></math></span> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn><mo>/</mo><mn>3</mn></math></span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>4</mn><mo>/</mo><mn>3</mn></math></span> </div>

answer:
<div class="solution"><p>We have,</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>+</mo><mi>y</mi><mo>+</mo><mi>z</mi><mo>=</mo><mn>5</mn></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><msup><mi>y</mi><mn>2</mn></msup><mo>+</mo><msup><mi>z</mi><mn>2</mn></msup><mo>=</mo><mn>9</mn></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mo> </mo><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><msup><mi>y</mi><mn>2</mn></msup><mo>+</mo><mo>(</mo><mn>5</mn><mo>-</mo><mi>x</mi><mo>-</mo><mi>y</mi><msup><mo>)</mo><mn>2</mn></msup><mo>=</mo><mn>9</mn></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mo> </mo><msup><mi>y</mi><mn>2</mn></msup><mo>+</mo><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mn>5</mn><mi>x</mi><mo>-</mo><mn>5</mn><mi>y</mi><mo>+</mo><mi>x</mi><mi>y</mi><mo>+</mo><mn>8</mn><mo>=</mo><mn>0</mn></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mo> </mo><msup><mi>y</mi><mn>2</mn></msup><mo>+</mo><mi>y</mi><mo>(</mo><mi>x</mi><mo>-</mo><mn>5</mn><mo>)</mo><mo>+</mo><mfenced><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mn>5</mn><mi>x</mi><mo>+</mo><mn>8</mn></mrow></mfenced><mo>=</mo><mn>0</mn></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo></math> Since <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>∈</mo><mi>R</mi></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mi>x</mi><mo>-</mo><mn>5</mn><msup><mo>)</mo><mn>2</mn></msup><mo>-</mo><mn>4</mn><mfenced><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mn>5</mn><mi>x</mi><mo>+</mo><mn>8</mn></mrow></mfenced><mo>≥</mo><mn>0</mn></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mo> </mo><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mn>10</mn><mi>x</mi><mo>+</mo><mn>25</mn><mo>-</mo><mn>4</mn><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>20</mn><mi>x</mi><mo>-</mo><mn>32</mn><mo>≥</mo><mn>0</mn></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mo> </mo><mn>3</mn><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mn>10</mn><mi>x</mi><mo>+</mo><mn>7</mn><mo>≤</mo><mn>0</mn></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mo> </mo><mn>3</mn><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mn>7</mn><mi>x</mi><mo>-</mo><mn>3</mn><mi>x</mi><mo>+</mo><mn>7</mn><mo>≤</mo><mn>0</mn></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mo> </mo><mo>(</mo><mn>3</mn><mi>x</mi><mo>-</mo><mn>7</mn><mo>)</mo><mo>(</mo><mi>x</mi><mo>-</mo><mn>1</mn><mo>)</mo><mo>≤</mo><mn>0</mn></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mo> </mo><mi>x</mi><mo>∈</mo><mfenced close="]" open="["><mrow><mn>1</mn><mo>,</mo><mfrac><mn>7</mn><mn>3</mn></mfrac></mrow></mfenced></math><br/>Length of interval <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfenced><mrow><mfrac><mn>7</mn><mn>3</mn></mfrac><mo>-</mo><mn>1</mn></mrow></mfenced><mo>=</mo><mfrac><mn>4</mn><mn>3</mn></mfrac></math></p></div>


title: if-y-1-c-o-s-2-1-c-o-s-2-then-d-d-y-at-4-3-is
slug: if-y-1-c-o-s-2-1-c-o-s-2-then-d-d-y-at-4-3-is-59550
tags: ['Mathematics', 'Differentiation', 'BITSAT', 'BITSAT 2023 (Memory Based Paper 1)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">-2</span> </div>
source_db: MarksBatch2_P1.db
question:
<div class="question">If $y=\sqrt{\left(\frac{1+\cos 2 \theta}{1-\cos 2 \theta}\right)}$, then $\frac{d y}{d \theta}$ at $\theta=\frac{3 \pi}{4}$ is :</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">-2</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">2</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\pm 2$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">None of these</span> </li> </ul>

answer:
<div class="solution">$y=\sqrt{\frac{1+\cos 2 \theta}{1-\cos 2 \theta}}$<br/>$\Rightarrow y=\sqrt{\frac{2 \cos ^2 \theta}{2 \sin ^2 \theta}}=\sqrt{\cot ^2 \theta}$<br/>$\Rightarrow y=\cot \theta$<br/>Differentiate w.r.t. ' $\theta$ ', we get: $\frac{d y}{d \theta}=-\operatorname{cosec}^2 \theta$<br/>Now, $\left(\frac{\mathrm{dy}}{\mathrm{d} \theta}\right)_{\theta=\frac{3 \pi}{4}}=-\operatorname{cosec}^2\left(\frac{3 \pi}{4}\right)$<br/>$=-\operatorname{cosec}^2\left(\pi-\frac{\pi}{4}\right)=-\operatorname{cosec}^2 \frac{\pi}{4}=-2$</div>


title: if-y-1-c-o-s-s-i-n-2-s-i-n-then-value-of-1-s-i-n-1-c-o-s-s-i-n-is
slug: if-y-1-c-o-s-s-i-n-2-s-i-n-then-value-of-1-s-i-n-1-c-o-s-s-i-n-is-17461
tags: ['Mathematics', 'Trigonometric Equations', 'BITSAT', 'BITSAT 2021']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">$y$</span> </div>
source_db: MarksBatch2_P1.db
question:
<div class="question">If $y=\frac{2 \sin \alpha}{1+\cos \alpha+\sin \alpha}$, then value of $\frac{1-\cos \alpha+\sin \alpha}{1+\sin \alpha}$ is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\frac{y}{3}$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$y$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$2 y$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\frac{3}{2} y$</span> </li> </ul>

answer:
<div class="solution">$\frac{1-\cos \alpha+\sin \alpha}{1+\sin \alpha}$
<br/> <br/>$=\frac{1-\cos \alpha+\sin \alpha}{1+\sin \alpha} \cdot \frac{1+\cos \alpha+\sin \alpha}{1+\cos \alpha+\sin \alpha}$
<br/> <br/>$=\frac{(1+\sin \alpha)^{2}-\cos ^{2} \alpha}{(1+\sin \alpha)(1+\cos \alpha+\sin \alpha)}$
<br/> <br/>$=\frac{\left(1+\sin ^{2} \alpha+2 \sin \alpha\right)-\left(1-\sin ^{2} \alpha\right)}{(1+\sin \alpha)(1+\cos \alpha+\sin \alpha)}$
<br/> <br/>$=\frac{2 \sin \alpha}{1+\cos \alpha+\sin \alpha}=y$</div>


title: if-y-1-x-2-x-2-3-x-3-then-d-x-d-y-is-equal-to
slug: if-y-1-x-2-x-2-3-x-3-then-d-x-d-y-is-equal-to-85445
tags: ['Mathematics', 'Differentiation', 'BITSAT', 'BITSAT 2013']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">$y$</span> </div>
answer: <div class="solution">$y=e^{x} \Rightarrow \frac{d y}{d x}=e^{x}=y$</div>
source_db: MarksBatch2_P1.db
question:
<div class="question">If $y=1+x+\frac{x^{2}}{2 !}+\frac{x^{3}}{3 !}+\ldots . \infty,$ then $\frac{d y}{d x}$ is equal to</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$x$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">1</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$y$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">None of these</span> </li> </ul>


title: if-y-3-x-6-x-2-10-x-3-then-3-1-y-3-2-2-14-y-2-3-2-3-147-y-3-is-equal-to
slug: if-y-3-x-6-x-2-10-x-3-then-3-1-y-3-2-2-14-y-2-3-2-3-147-y-3-is-equal-to-12792
tags: ['Mathematics', 'Sequences and Series', 'BITSAT', 'BITSAT 2020']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">$x$</span> </div>
source_db: MarksBatch2_P1.db
question:
<div class="question">If $y=3 x+6 x^{2}+10 x^{3}+\ldots \ldots \ldots, \infty,$ then
<br/> <br/>$\frac{1}{3} y-\frac{1.4}{3^{2} 2} y^{2}+\frac{1.4 .7}{3^{2} 3} y^{3}-\ldots \ldots \infty$ is equal to</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$x$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$1-x$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$1+x$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$x^{x}$</span> </li> </ul>

answer:
<div class="solution">We have $y=3 x+6 x^{2}+10 x^{3}+\ldots \ldots$
<br/> <br/>$\Rightarrow 1+y=1+3 x+6 x^{2}+10 x^{3}+\ldots$
<br/> <br/>$\Rightarrow 1+y=(1-x)^{-3} \Rightarrow 1-x=(1+y)^{-1 / 3}$
<br/> <br/>$\Rightarrow x=1-(1+y)^{-1 / 3}$
<br/> <br/>$=\frac{1}{3} y-\frac{1.4}{3^{2} \cdot 2} y^{2}+\frac{1 \cdot 4 \cdot 7}{3^{2} \cdot 3} y^{3}-\ldots$</div>


title: if-y-x-1-x-2-n-then-1-x-2-d-x-2-d-2-y-x-d-x-d-y-is-1
slug: if-y-x-1-x-2-n-then-1-x-2-d-x-2-d-2-y-x-d-x-d-y-is-1-30587
tags: ['Mathematics', 'Differential Equations', 'BITSAT', 'BITSAT 2015']
source_db: MarksBatch2_P1.db
question:
<div class="question">If $y=\left(x+\sqrt{1+x^{2}}\right)^{n},$ then $\left(1+x^{2}\right) \frac{d^{2} y}{d x^{2}}+x \frac{d y}{d x}$ is</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$n^{2} y$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$-n^{2} y$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$-y$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$2 x^{2} y$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$n^{2} y$</span> </div>

answer:
<div class="solution">$y=\left(x+\sqrt{1+x^{2}}\right)^{n}$
<br/> <br/>$$
<br/> <br/>\begin{array}{c}
<br/> <br/>\frac{d y}{d x}=n\left(x+\sqrt{1+x^{2}}\right)^{n-1}\left(1+\frac{1}{2}\left(1+x^{2}\right)^{-1 / 2} \cdot 2 x\right) \\
<br/> <br/>\qquad \begin{array}{c}
<br/> <br/>\frac{d y}{d x}=n\left(x+\sqrt{1+x^{2}}\right)^{n-1} \frac{\left(\sqrt{1+x^{2}}+x\right)}{\sqrt{1+x^{2}}} \\
<br/> <br/>=\frac{n\left(\sqrt{1+x^{2}}+x\right)^{n}}{\sqrt{1+x^{2}}}
<br/> <br/>\end{array}
<br/> <br/>\end{array}
<br/> <br/>$$
<br/> <br/>or $\sqrt{1+x^{2}} \frac{d y}{d x}=n y$ or $\sqrt{1+x^{2}} y_{1}=n y$
<br/> <br/>$$
<br/> <br/>\left(y_{1}=\frac{d y}{d x}\right)
<br/> <br/>$$
<br/> <br/>Squaring, $\left(1+x^{2}\right) y_{1}^{2}=n^{2} y^{2}$
<br/> <br/>Differentiating,
<br/> <br/>$$
<br/> <br/>\left(1+x^{2}\right) 2 y_{1} y_{2}+y_{1}^{2} \cdot 2 x=n^{2} \cdot 2 y y_{1}
<br/> <br/>$$
<br/> <br/>or $\left(1+x^{2}\right) y_{2}+x y_{1}=n^{2} y$</div>


title: if-y-x-e-y-x-then-d-x-d-y-is-equal-to
slug: if-y-x-e-y-x-then-d-x-d-y-is-equal-to-27741
question: <div class="question">If $y^x=e^{y-x}$, then $\frac{d y}{d x}$ is equal to</div>
tags: ['Mathematics', 'Differentiation', 'BITSAT', 'BITSAT 2023 (Memory Based Paper 2)']
source_db: MarksBatch2_P1.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\frac{1+\log y}{y \log y}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\frac{(1+\log y)^2}{y \log y}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\frac{1+\log y}{(\log y)^2}$</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">$\frac{(1+\log y)^2}{\log y}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\frac{(1+\log y)^2}{\log y}$</span> </div>

answer:
<div class="solution">Here, $\mathrm{y}^{\mathrm{x}}=\mathrm{e}^{\mathrm{y}-\mathrm{x}}$<br/>Taking log on both sides, we get<br/>$\begin{aligned}<br/>&amp; \log y^x=\log e^{y-x} \\<br/>&amp; \quad\left(\because \log a^b=b \log a \text { and } \log e=1\right)<br/>\end{aligned}$<br/>$\Rightarrow x \log y=(y-x) \log e \Rightarrow x \log y=y-x \ldots(i)$<br/>On differentiating w.r.t. $x$, we get<br/>$\frac{d}{d x}(x \log y)=\frac{d}{d x}(y-x)$<br/>(using product rule)<br/>$\begin{aligned}<br/>&amp; \Rightarrow x\left(\frac{1}{y}\right) \frac{d y}{d x}+\log y(1)=\frac{d y}{d x}-1 \\<br/>&amp; \Rightarrow \frac{d y}{d x}\left(\frac{x}{y}-1\right)=-1-\log y \\<br/>&amp; \Rightarrow \frac{d y}{d x}\left[\frac{y}{(1+\log y) y}-1\right]=-(1+\log y) \\<br/>&amp; \qquad \quad\left[\because \text { from eq. }(i) \cdot x=\frac{y}{(1+\log y)}\right] \\<br/>&amp; \Rightarrow \frac{d y}{d x}\left[\frac{1-1-\log y}{1+\log y}\right]=-(1+\log y)<br/>\end{aligned}$<br/>$\Rightarrow \frac{d y}{d x}=-\frac{(1+\log y)^2}{-\log y}$<br/>$\Rightarrow \frac{d y}{d x}=\frac{(1+\log y)^2}{\log y}$</div>


title: if-yex-cos-x-and-y4k-y0-where-y4fracd4-yd-x4-then-k
slug: if-yex-cos-x-and-y4k-y0-where-y4fracd4-yd-x4-then-k-60515
tags: ['Mathematics', 'Functions', 'BITSAT', 'BITSAT 2009']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">4</span> </div>
source_db: MarksBatch2_P1.db
question:
<div class="question">If \(y=e^{-x} \cos x\) and \(y_4+k y=0\), where \(y_4=\frac{d^4 y}{d x^4}\), then \(k=\)</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">4</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">-4</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">2</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">-2</span> </li> </ul>

answer:
<div class="solution">Let \(y=e^{-x} \cos x\)<br/>\(\begin{aligned}<br/>&amp; \mathrm{y}_1=-e^{-x} \sin x-e^{-x} \cos x=-e^{-x} \sin x-y \\<br/>&amp; y_2=-e^{-x} \cos x+e^{-x} \sin x-y_1 \\<br/>&amp; \Rightarrow y_2=-y-y_1+e^{-x} \sin x=-2\left(y+y_1\right) \\<br/>&amp; \Rightarrow y_3=-2\left(y_1+y_2\right)=-2\left(e^{-x} \sin x-y\right) \\<br/>&amp; \Rightarrow y_4=4 y_1+2 y_2=4 y_1-4 y-4 y_1 \text { or } y_4+4 y=0 \\<br/>&amp; \Rightarrow k=4<br/>\end{aligned}\)</div>


title: if-yfrac2-sin-alpha1cos-alphasin-alpha-then-value-of-frac1cos-alphasin-alpha1sin-alpha-is
slug: if-yfrac2-sin-alpha1cos-alphasin-alpha-then-value-of-frac1cos-alphasin-alpha1sin-alpha-is-12602
tags: ['Mathematics', 'Trigonometric Equations', 'BITSAT', 'BITSAT 2009']
source_db: MarksBatch2_P1.db
question:
<div class="question">If \(y=\frac{2 \sin \alpha}{1+\cos \alpha+\sin \alpha}\), then value of \(\frac{1-\cos \alpha+\sin \alpha}{1+\sin \alpha}\) is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">\(\frac{y}{3}\)</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">\(\mathrm{y}\)</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">\(2 y\)</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">\(\frac{3}{2} y\)</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">\(\mathrm{y}\)</span> </div>

answer:
<div class="solution">\(\begin{aligned}<br/>&amp; \frac{1-\cos \alpha+\sin \alpha}{1+\sin \alpha} \\<br/>&amp; =\frac{1-\cos \alpha+\sin \alpha}{1+\sin \alpha} \cdot \frac{1+\cos \alpha+\sin \alpha}{1+\cos \alpha+\sin \alpha} \\<br/>&amp; =\frac{(1+\sin \alpha)^2-\cos ^2 \alpha}{(1+\sin \alpha)(1+\cos \alpha+\sin \alpha)} \\<br/>&amp; =\frac{2 \sin \alpha(1+\sin \alpha)}{(1+\sin \alpha)(1+\cos \alpha+\sin \alpha)} \\<br/>&amp; =\frac{2 \sin \alpha}{1+\cos \alpha+\sin \alpha}=y<br/>\end{aligned}\)</div>


title: if-yxx2-then-fracd-yd-x-is-equal-to
slug: if-yxx2-then-fracd-yd-x-is-equal-to-26498
question: <div class="question">If \(y=x^{x^2}\), then \(\frac{d y}{d x}\) is equal to</div>
tags: ['Mathematics', 'Differentiation', 'BITSAT', 'BITSAT 2011']
source_db: MarksBatch2_P1.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">\((2 \ln x)\)</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">\((2 \ln x+1)\)</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">\((\ln \ln x+1) x^{x^2}\)</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">None of these</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

answer:
<div class="solution">Consider \(y=x^{x^2} \Rightarrow \ln y=x^2 \ln x\)<br/>\(\begin{aligned}<br/>&amp; \frac{1}{y} \frac{d y}{d x}=2 x \ln x+x^2 \cdot \frac{1}{x}=x(1+2 \ln x) \\<br/>&amp; \frac{d y}{d x}=x^{x^2} \cdot x(1+2 \ln x)=x^{x^2+1}(1+2 \ln x)<br/>\end{aligned}\)</div>


title: if-z-1-2-cos-4-i-sin-4-and-z-2-3-cos-3-i-sin-3-then-z-1-z-2-is-equal-to-m-value-of-m-is
slug: if-z-1-2-cos-4-i-sin-4-and-z-2-3-cos-3-i-sin-3-then-z-1-z-2-is-equal-to-m-value-of-m-is-83246
tags: ['Mathematics', 'Complex Number', 'BITSAT', 'BITSAT 2021']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">6</span> </div>
source_db: MarksBatch2_P1.db
question:
<div class="question">$\text { If } z_{1}=\sqrt{2}\left[\cos \frac{\pi}{4}+i \sin \frac{\pi}{4}\right] \text { and } z_{2}=\sqrt{3}\left[\cos \frac{\pi}{3}+i \sin \frac{\pi}{3}\right]$ then $\left|z_{1} z_{2}\right|$ is equal to $\sqrt{m}$. Value of $m$ is</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">6</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">3</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">2</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">5</span> </li> </ul>

answer:
<div class="solution">$z_{1}=\sqrt{2}\left[\cos \frac{\pi}{4}+i \sin \frac{\pi}{4}\right]=\sqrt{2}\left[\frac{1}{\sqrt{2}}+i \frac{1}{\sqrt{2}}\right]=1+i$
<br/> <br/>$\left|z_{1}\right|=\sqrt{2}$
<br/> <br/>and $z_{2}=\sqrt{3}\left[\cos \frac{\pi}{3}+i \sin \frac{\pi}{3}\right]=\sqrt{3}\left[\frac{1}{2}+i \frac{\sqrt{3}}{2}\right]$
<br/> <br/>$\left|z_{2}\right|=\sqrt{\frac{3}{4}+\frac{9}{4}}=\sqrt{3}$
<br/> <br/>$\left|z_{1} z_{2}\right|=\left|z_{1}\right|\left|z_{2}\right|=\sqrt{2} \cdot \sqrt{3}=6$</div>


title: if-z-1-3-i-3-and-z-2-3-i-then-the-complex-number-z-2-z-1-50-lies-in-the
slug: if-z-1-3-i-3-and-z-2-3-i-then-the-complex-number-z-2-z-1-50-lies-in-the-31750
tags: ['Mathematics', 'Complex Number', 'BITSAT', 'BITSAT 2016']
source_db: MarksBatch2_P1.db
question:
<div class="question">If $z_{1}=\sqrt{3}+i \sqrt{3}$ and $z_{2}=\sqrt{3}+i$, then the
<br/> <br/>complex number $\left(\frac{\mathrm{z}_{1}}{\mathrm{z}_{2}}\right)^{50}$ lies in the :</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">first quadrant</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">second quadrant</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">third quadrant</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">fourth quadrant</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">first quadrant</span> </div>

answer:
<div class="solution">$\left(\frac{z_{1}}{z_{2}}\right)^{50}=\left(\frac{\sqrt{3}+i \sqrt{3}}{\sqrt{3}+i}\right)^{50}$
<br/> <br/>$=\left[\left(\frac{\sqrt{3}(1+i)}{\sqrt{3}+i}\right)^{2}\right]^{25}=\left[\frac{3(2 i)}{3-1+2 \sqrt{3} i}\right]^{25}$
<br/> <br/>$=\left(\frac{3 i}{1+\sqrt{3} i}\right)^{25}=\frac{3^{25} i^{25}}{\left(-2 \omega^{2}\right)^{25}}$
<br/> <br/>$$
<br/> <br/>\begin{array}{l}
<br/> <br/>=-i . \omega\left(\frac{3}{2}\right)^{25}=-\mathrm{i}\left(\frac{-1+\sqrt{3} \mathrm{i}}{2}\right)\left(\frac{3}{2}\right)^{25} \\
<br/> <br/>=\left(\frac{3}{2}\right)^{25}\left(\frac{\sqrt{3}}{2}+\frac{1}{2} \mathrm{i}\right)
<br/> <br/>\end{array}
<br/> <br/>$$
<br/> <br/>Hence, $\left(\frac{z_{1}}{z_{2}}\right)^{50}$ lies in the first quadrant as both real and imaginary parts of this number are positive.</div>


tags: ['Mathematics', 'Complex Number', 'BITSAT', 'BITSAT 2019']
source_db: MarksBatch2_P1.db
title:
if-z-1-and-z-1-represents-adjacent-vertices-of-a-regular-polygon-of-n-sides-and-if-i-m-z-1-r-e-z-1-2-1-then-n-is-equal-to

slug:
if-z-1-and-z-1-represents-adjacent-vertices-of-a-regular-polygon-of-n-sides-and-if-i-m-z-1-r-e-z-1-2-1-then-n-is-equal-to-27887

question:
<div class="question">If <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>z</mi><mn>1</mn></msub></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mover><mi>z</mi><mo>¯</mo></mover><mn>1</mn></msub></math> represents adjacent vertices of a regular polygon of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi></math> sides and if <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mi>I</mi><mi>m</mi><mfenced><msub><mi>z</mi><mn>1</mn></msub></mfenced></mrow><mrow><mi>R</mi><mi>e</mi><mfenced><msub><mi>z</mi><mn>1</mn></msub></mfenced></mrow></mfrac><mo>=</mo><msqrt><mn>2</mn></msqrt><mo>-</mo><mn>1</mn></math>, then <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi></math> is equal to</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>8</mn></math></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>16</mn></math></span> </li><li class=""> <span class="option-label">C</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>18</mn></math></span> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>24</mn></math></span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>8</mn></math></span> </div>

answer:
<div class="solution"><p>Let <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>z</mi><mn>1</mn></msub><mo>=</mo><mi>r</mi><msup><mi>e</mi><mrow><mi>i</mi><mi>θ</mi></mrow></msup></math></p><p>then, <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mover><mi>z</mi><mo>¯</mo></mover><mn>1</mn></msub><mo>=</mo><mi>r</mi><msup><mi>e</mi><mrow><mo>-</mo><mi>i</mi><mi>θ</mi></mrow></msup></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><msub><mi>z</mi><mn>1</mn></msub><msub><mover><mi>z</mi><mo>¯</mo></mover><mn>1</mn></msub></mfrac><mo>=</mo><msup><mi>e</mi><mrow><mn>2</mn><mi>i</mi><mi>θ</mi></mrow></msup><mo>=</mo><msup><mi>e</mi><mrow><mn>2</mn><mi>π</mi><mi>i</mi><mo>/</mo><mi>n</mi></mrow></msup></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mo> </mo><mi>θ</mi><mo>=</mo><mfrac><mi>π</mi><mi>n</mi></mfrac></math></p><p>Also, <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><msub><mi>I</mi><mi>m</mi></msub><mfenced><msub><mi>z</mi><mn>1</mn></msub></mfenced></mrow><mrow><msub><mi>R</mi><mi>e</mi></msub><mfenced><msub><mi>z</mi><mn>1</mn></msub></mfenced></mrow></mfrac><mo>=</mo><mi>tan</mi><mi>θ</mi><mo>=</mo><mi>tan</mi><mfrac><mi>π</mi><mi>n</mi></mfrac><mo>=</mo><msqrt><mn>2</mn></msqrt><mo>-</mo><mn>1</mn></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mo> </mo><mi>tan</mi><mfrac><mi>π</mi><mn>8</mn></mfrac><mo>=</mo><msqrt><mn>2</mn></msqrt><mo>-</mo><mn>1</mn></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>∴</mo><mo> </mo><mi>n</mi><mo>=</mo><mn>8</mn></math></p></div>


title: if-z-1-z-2-and-z-3-represent-the-vertices-of-an-equilateral-triangle-such-that-z-1-z-2-z-3-then
tags: ['Mathematics', 'Complex Number', 'BITSAT', 'BITSAT 2017']
source_db: MarksBatch2_P1.db
slug:
if-z-1-z-2-and-z-3-represent-the-vertices-of-an-equilateral-triangle-such-that-z-1-z-2-z-3-then-18247

question:
<div class="question">If <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>z</mi><mn>1</mn></msub><mo>,</mo><msub><mi>z</mi><mn>2</mn></msub></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>z</mi><mn>3</mn></msub></math> represent the vertices of an equilateral triangle such that <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced close="|" open="|"><msub><mi>z</mi><mn>1</mn></msub></mfenced><mo>=</mo><mfenced close="|" open="|"><msub><mi>z</mi><mn>2</mn></msub></mfenced><mo>=</mo><mfenced close="|" open="|"><msub><mi>z</mi><mn>3</mn></msub></mfenced></math> then</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>z</mi><mn>1</mn></msub><mo>+</mo><msub><mi>z</mi><mn>2</mn></msub><mo>=</mo><msub><mi>z</mi><mn>3</mn></msub></math></span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>z</mi><mn>1</mn></msub><mo>+</mo><msub><mi>z</mi><mn>2</mn></msub><mo>+</mo><msub><mi>z</mi><mn>3</mn></msub><mo>=</mo><mn>0</mn></math></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>z</mi><mn>1</mn></msub><msub><mi>z</mi><mn>2</mn></msub><mo>=</mo><mfrac><mn>1</mn><msub><mi>z</mi><mn>3</mn></msub></mfrac></math></span> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>z</mi><mn>1</mn></msub><mo>-</mo><msub><mi>z</mi><mn>2</mn></msub><mo>=</mo><msub><mi>z</mi><mn>3</mn></msub><mo>-</mo><msub><mi>z</mi><mn>2</mn></msub></math></span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>z</mi><mn>1</mn></msub><mo>+</mo><msub><mi>z</mi><mn>2</mn></msub><mo>+</mo><msub><mi>z</mi><mn>3</mn></msub><mo>=</mo><mn>0</mn></math></span> </div>

answer:
<div class="solution"><blockquote><p>Since, <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced close="|" open="|"><msub><mi>z</mi><mn>1</mn></msub></mfenced><mo>=</mo><mfenced close="|" open="|"><msub><mi>z</mi><mn>2</mn></msub></mfenced><mo>=</mo><mfenced close="|" open="|"><msub><mi>z</mi><mn>3</mn></msub></mfenced><mo>.</mo></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mfenced close="|" open="|"><mrow><msub><mi>z</mi><mn>1</mn></msub><mo>-</mo><mn>0</mn></mrow></mfenced><mo>=</mo><mfenced close="|" open="|"><mrow><msub><mi>z</mi><mn>2</mn></msub><mo>-</mo><mn>0</mn></mrow></mfenced><mo>=</mo><mfenced close="|" open="|"><mrow><msub><mi>z</mi><mn>3</mn></msub><mo>-</mo><mn>0</mn></mrow></mfenced><mo>.</mo></math></p><p>Thus, the given points are equidistant from origin, and we know that the point which is equidistant from the vertices of a triangle is the circumcentre of the triangle.</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>∴</mo><mo> </mo><mn>0</mn></math> is the circumcentre of an equilateral <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>∆</mo><mi>A</mi><mi>B</mi><mi>C</mi></math> and we also know that for an equilateral triangle the centroid and circumcentre are same.</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>∴</mo><mo> </mo><mfrac><mrow><msub><mi>x</mi><mn>1</mn></msub><mo>+</mo><msub><mi>x</mi><mn>2</mn></msub><mo>+</mo><msub><mi>x</mi><mn>3</mn></msub></mrow><mn>3</mn></mfrac><mo>=</mo><mn>0</mn><mo>=</mo><mfrac><mrow><msub><mi>y</mi><mn>1</mn></msub><mo>+</mo><msub><mi>y</mi><mn>2</mn></msub><mo>+</mo><msub><mi>y</mi><mn>3</mn></msub></mrow><mn>3</mn></mfrac></math></p><p>where, <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>z</mi><mn>1</mn></msub><mo>=</mo><msub><mi>x</mi><mn>1</mn></msub><mo>+</mo><mi>i</mi><msub><mi>y</mi><mn>1</mn></msub></math>.</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mfrac><mrow><msub><mi>x</mi><mn>1</mn></msub><mo>+</mo><msub><mi>x</mi><mn>2</mn></msub><mo>+</mo><msub><mi>x</mi><mn>3</mn></msub></mrow><mn>3</mn></mfrac><mo>+</mo><mi>i</mi><mfenced><mfrac><mrow><msub><mi>y</mi><mn>1</mn></msub><mo>+</mo><msub><mi>y</mi><mn>2</mn></msub><mo>+</mo><msub><mi>y</mi><mn>3</mn></msub></mrow><mn>3</mn></mfrac></mfenced><mo>=</mo><mn>0</mn></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mfenced><mrow><msub><mi>x</mi><mn>1</mn></msub><mo>+</mo><mi>i</mi><msub><mi>y</mi><mn>1</mn></msub></mrow></mfenced><mo>+</mo><mfenced><mrow><msub><mi>x</mi><mn>2</mn></msub><mo>+</mo><mi>i</mi><msub><mi>y</mi><mn>2</mn></msub></mrow></mfenced><mo>+</mo><mfenced><mrow><msub><mi>x</mi><mn>3</mn></msub><mo>+</mo><mi>i</mi><msub><mi>y</mi><mn>3</mn></msub></mrow></mfenced><mo>=</mo><mn>0</mn></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><msub><mi>z</mi><mn>1</mn></msub><mo>+</mo><msub><mi>z</mi><mn>2</mn></msub><mo>+</mo><msub><mi>z</mi><mn>3</mn></msub><mo>=</mo><mn>0</mn><mo>.</mo></math></p></blockquote></div>


title: if-z-1-z-2-z-n-1-then-the-value-of-z-1-z-2-z-n-z-1-1-z-2-1-z-n-1-is
slug: if-z-1-z-2-z-n-1-then-the-value-of-z-1-z-2-z-n-z-1-1-z-2-1-z-n-1-is-72644
tags: ['Mathematics', 'Complex Number', 'BITSAT', 'BITSAT 2020']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">0</span> </div>
source_db: MarksBatch2_P1.db
question:
<div class="question">If $\left|z_{1}\right|=\left|z_{2}\right|=\ldots \ldots \ldots . .\left|z_{n}\right|=1,$ then the
<br/> <br/>value of $\left|z_{1}+z_{2}+\ldots \ldots . z_{n}\right|-$
<br/> <br/>$\left|\frac{1}{z_{1}}+\frac{1}{z_{2}}+\ldots \ldots .+\frac{1}{z_{n}}\right|$ is,</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">0</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">1</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">-1</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">None</span> </li> </ul>

answer:
<div class="solution">$z_{1} \bar{z}_{1}=z_{2} \bar{z}_{2}=\ldots .=z_{n} \bar{z}_{n}=1$
<br/> <br/>$\Rightarrow \overline{\mathrm{z}}_{1}=\frac{1}{\mathrm{z}_{1}}, \overline{\mathrm{z}}_{2}=\frac{1}{\mathrm{z}_{2}}, \overline{\mathrm{z}}_{3}=\frac{1}{\mathrm{z}_{3}}, \ldots \ldots, \overline{\mathrm{z}}_{\mathrm{n}}=\frac{1}{\mathrm{z}_{\mathrm{n}}}$
<br/> <br/>$\therefore\left|z_{1}+z_{2}+\ldots .+z_{n}\right|-\left|\frac{1}{z_{1}}+\frac{1}{z_{2}}+\ldots .+\frac{1}{z_{n}}\right|$
<br/> <br/>$=\left|z_{1}+z_{2}+\ldots+z_{n}\right|-\left|\bar{z}_{1}+\bar{z}_{2}+\ldots .+\bar{z}_{n}\right|=0$</div>


title: if-z-a-i-b-satisfies-arg-z-1-arg-z-3-i-then-a-1-b
slug: if-z-a-i-b-satisfies-arg-z-1-arg-z-3-i-then-a-1-b-65751
tags: ['Mathematics', 'Complex Number', 'BITSAT', 'BITSAT 2017']
source_db: MarksBatch2_P1.db
question:
<div class="question">If <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>z</mi><mo>=</mo><mi>a</mi><mo>+</mo><mi>i</mi><mi>b</mi></math> satisfies <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>arg</mi><mo>(</mo><mi>z</mi><mo>-</mo><mn>1</mn><mo>)</mo><mo>=</mo><mi>arg</mi><mo>(</mo><mi>z</mi><mo>+</mo><mn>3</mn><mi>i</mi><mo>)</mo></math>, then <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mi>a</mi><mo>-</mo><mn>1</mn><mo>)</mo><mo>:</mo><mi>b</mi><mo>=</mo></math></div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><mo>:</mo><mn>1</mn></math></span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn><mo>:</mo><mn>3</mn></math></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mn>1</mn><mo>:</mo><mn>3</mn></math></span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">None of these</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn><mo>:</mo><mn>3</mn></math></span> </div>

answer:
<div class="solution"><p>We have,</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>arg</mi><mo>(</mo><mi>z</mi><mo>-</mo><mn>1</mn><mo>)</mo><mo>=</mo><mi>arg</mi><mo>(</mo><mi>z</mi><mo>+</mo><mn>3</mn><mi>i</mi><mo>)</mo></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mi>arg</mi><mo>(</mo><mo>(</mo><mi>a</mi><mo>-</mo><mn>1</mn><mo>)</mo><mo>+</mo><mi>i</mi><mi>b</mi><mo>)</mo><mo>=</mo><mi>arg</mi><mo>(</mo><mi>a</mi><mo>+</mo><mo>(</mo><mi>b</mi><mo>+</mo><mn>3</mn><mo>)</mo><mi>i</mi><mo>)</mo></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><msup><mi>tan</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup><mfenced><mfrac><mi>b</mi><mrow><mi>a</mi><mo>-</mo><mn>1</mn></mrow></mfrac></mfenced><mo>=</mo><msup><mi>tan</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup><mfenced><mfrac><mrow><mi>b</mi><mo>+</mo><mn>3</mn></mrow><mi>a</mi></mfrac></mfenced></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mo> </mo><mfrac><mi>b</mi><mrow><mi>a</mi><mo>-</mo><mn>1</mn></mrow></mfrac><mo>=</mo><mfrac><mrow><mi>b</mi><mo>+</mo><mn>3</mn></mrow><mi>a</mi></mfrac></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mo> </mo><mi>a</mi><mi>b</mi><mo>=</mo><mo>(</mo><mi>a</mi><mo>-</mo><mn>1</mn><mo>)</mo><mo>(</mo><mi>b</mi><mo>+</mo><mn>3</mn><mo>)</mo></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mo> </mo><mn>3</mn><mo>(</mo><mi>a</mi><mo>-</mo><mn>1</mn><mo>)</mo><mo>=</mo><mi>b</mi></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mo> </mo><mo>(</mo><mi>a</mi><mo>-</mo><mn>1</mn><mo>)</mo><mo>:</mo><mi>b</mi><mo>=</mo><mn>1</mn><mo>:</mo><mn>3</mn></math></p></div>


title: if-z-is-a-complex-number-such-that-z-2-z-2-then
slug: if-z-is-a-complex-number-such-that-z-2-z-2-then-93370
question: <div class="question">If $z$ is a complex number such that $z^{2}=(\bar{z})^{2}$, then</div>
tags: ['Mathematics', 'Complex Number', 'BITSAT', 'BITSAT 2021']
source_db: MarksBatch2_P1.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$z$ is purely real</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$z$ is purely imaginary</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">either $z$ is purely real or purely imaginary</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">None of these</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">either $z$ is purely real or purely imaginary</span> </div>

answer:
<div class="solution">Let $z=x+i y$, then its conjugate $\bar{z}=x-i y$
<br/> <br/>Given that $z^{2}=(\bar{z})^{2}$
<br/> <br/>$\begin{array}{l}
<br/> <br/>\Rightarrow x^{2}-y^{2}+2 i x y=x^{2}-y^{2}-2 i x y \\
<br/> <br/>\Rightarrow 4 i x y=0
<br/> <br/>\end{array}$</div>


title: if-z-x-i-y-z-13-a-ib-then-a-x-b-y-k-a-2-b-2-where-k-is-equal-to
slug: if-z-x-i-y-z-13-a-ib-then-a-x-b-y-k-a-2-b-2-where-k-is-equal-to-79084
tags: ['Mathematics', 'Complex Number', 'BITSAT', 'BITSAT 2014']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">4</span> </div>
source_db: MarksBatch2_P1.db
question:
<div class="question">If $z=x+i y, z^{1 / 3}=a-i b,$ then $\frac{x}{a}-\frac{y}{b}=k\left(a^{2}-b^{2}\right)$
<br/> <br/>where $k$ is equal to</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">1</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">2</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">3</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">4</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

answer:
<div class="solution">$z^{1 / 3}=a-i b \Rightarrow z=(a-i b)^{3}$
<br/> <br/>$\therefore x+i y=a^{3}+i b^{3}-3 i a^{2} b-3 a b^{2}$
<br/> <br/>$\Rightarrow x=a^{3}-3 a b^{2} \Rightarrow \frac{x}{a}=a^{2}-3 b^{2}$
<br/> <br/>and $y=b^{3}-3 a^{2} b \Rightarrow \frac{y}{b}=b^{2}-3 a^{2}$
<br/> <br/>So, $\frac{x}{a}-\frac{y}{b}=4\left(a^{2}-b^{2}\right)$</div>


title: ifm-d-is-12-the-value-of-sd-will-be
slug: ifm-d-is-12-the-value-of-sd-will-be-63493
question: <div class="question">IfM. D. is 12, the value of S.D. will be</div>
tags: ['Mathematics', 'Statistics', 'BITSAT', 'BITSAT 2014']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">15</span> </div>
source_db: MarksBatch2_P1.db
options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">15</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">12</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">24</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">None of these</span> </li> </ul>

answer:
<div class="solution">We know that Q.D $=\frac{5}{6} \times M \cdot D .=\frac{5}{6} \times 12=10$
<br/> <br/>$\therefore \mathrm{S} . \mathrm{D}=\frac{3}{2} \times Q \cdot D .=\frac{3}{2} \times 10 \Rightarrow S . D .=15$</div>


title: in-a-1-st-order-reaction-reactant-concentration-c-varies-with-time-t-as
slug: in-a-1-st-order-reaction-reactant-concentration-c-varies-with-time-t-as-24998
tags: ['Chemistry', 'Chemical Kinetics', 'BITSAT', 'BITSAT 2021']
source_db: MarksBatch2_P1.db
question:
<div class="question">In a 1 st order reaction, reactant concentration $C$ varies with time $t$ as :</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\frac{1}{\mathrm{C}}$ increases linearly with $t$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$\log \mathrm{C}$ decreases linearly with $t$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">C decreases with $\frac{1}{t}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\log \mathrm{C}$ decreases with $\frac{1}{t}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\log \mathrm{C}$ decreases linearly with $t$</span> </div>

answer:
<div class="solution"><img src="https://cdn-question-pool.getmarks.app/pyq/bitsat/IWh_hcrQ_xdvI4tyo60mF4132iU4ALvJ1PP3QDMWYLI.original.fullsize.png"/></div>


tags: ['Mathematics', 'Vector Algebra', 'BITSAT', 'BITSAT 2018']
source_db: MarksBatch2_P1.db
title:
in-a-a-b-c-d-e-f-are-the-midpoints-of-the-sides-b-c-c-a-and-a-b-respectively-the-vector-a-d-is-equal-to

slug:
in-a-a-b-c-d-e-f-are-the-midpoints-of-the-sides-b-c-c-a-and-a-b-respectively-the-vector-a-d-is-equal-to-22460

question:
<div class="question">In a <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>∆</mo><mi>A</mi><mi>B</mi><mi>C</mi><mo>,</mo></math> <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>D</mi><mo>,</mo><mo> </mo><mi>E</mi><mo>,</mo><mo> </mo><mi>F</mi></math> are the mid-points of the sides <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>B</mi><mi>C</mi></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>C</mi><mi>A</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi><mi>B</mi></math> respectively, the vector <math xmlns="http://www.w3.org/1998/Math/MathML"><mover><mrow><mi>A</mi><mi>D</mi></mrow><mo>→</mo></mover></math> is equal to</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mover><mrow><mi>B</mi><mi>E</mi></mrow><mo>→</mo></mover><mo>+</mo><mover><mrow><mi>C</mi><mi>F</mi></mrow><mo>→</mo></mover></math></span> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mover><mrow><mi>B</mi><mi>E</mi></mrow><mo>→</mo></mover><mo>-</mo><mover><mrow><mi>C</mi><mi>F</mi></mrow><mo>→</mo></mover></math></span> </li><li class=""> <span class="option-label">C</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mover><mrow><mi>C</mi><mi>F</mi></mrow><mo>→</mo></mover><mo>-</mo><mover><mrow><mi>B</mi><mi>E</mi></mrow><mo>→</mo></mover></math></span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mover><mrow><mi>B</mi><mi>E</mi></mrow><mo>→</mo></mover><mo>-</mo><mover><mrow><mi>C</mi><mi>F</mi></mrow><mo>→</mo></mover></math></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mover><mrow><mi>B</mi><mi>E</mi></mrow><mo>→</mo></mover><mo>-</mo><mover><mrow><mi>C</mi><mi>F</mi></mrow><mo>→</mo></mover></math></span> </div>

answer:
<div class="solution"><p><img src="https://cdn-question-pool.getmarks.app/pyq/bitsat/78f10128-ccbf-41c5-8ffa-86e5d1ef4ddc-d52.png" style="width: 343px; height: 200px;"/></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mover><mrow><mi>A</mi><mi>D</mi></mrow><mo>→</mo></mover><mo>=</mo><mn>3</mn><mover><mrow><mi>G</mi><mi>D</mi></mrow><mo>→</mo></mover></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>3</mn><mo>·</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mfenced><mrow><mover><mrow><mi>G</mi><mi>B</mi></mrow><mo>→</mo></mover><mo>+</mo><mover><mrow><mi>G</mi><mi>C</mi></mrow><mo>→</mo></mover></mrow></mfenced></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfrac><mn>3</mn><mn>2</mn></mfrac><mfenced><mrow><mfrac><mn>2</mn><mn>3</mn></mfrac><mover><mrow><mi>E</mi><mi>B</mi></mrow><mo>→</mo></mover><mo>+</mo><mfrac><mn>2</mn><mn>3</mn></mfrac><mover><mrow><mi>F</mi><mi>C</mi></mrow><mo>→</mo></mover></mrow></mfenced></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mo>-</mo><mover><mrow><mi>B</mi><mi>E</mi></mrow><mo>→</mo></mover><mo>-</mo><mover><mrow><mi>C</mi><mi>F</mi></mrow><mo>→</mo></mover></math></p><p>Hence, option (d) is correct.</p></div>


tags: ['Mathematics', 'Properties of Triangles', 'BITSAT', 'BITSAT 2015']
source_db: MarksBatch2_P1.db
title:
in-a-a-bc-the-lengths-of-the-two-larger-sides-are-10-and-9-units-respectively-if-the-angles-are-in-ap-then-the-length-of-the-third-side-can-be

slug:
in-a-a-bc-the-lengths-of-the-two-larger-sides-are-10-and-9-units-respectively-if-the-angles-are-in-ap-then-the-length-of-the-third-side-can-be-26199

question:
<div class="question">In a $\triangle A B C$, the lengths of the two larger sides are 10 and 9 units, respectively. If the angles are in $\mathrm{AP}$, then the length of the third side can be</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$5 \pm \sqrt{6}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$3 \sqrt{3}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\underline{5}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">None of these</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$5 \pm \sqrt{6}$</span> </div>

answer:
<div class="solution">Let A, B and $C$ be the three angles of $\Delta A B C$ and Let $a=10$ and $b=9$ It is given that the angles are in $\mathrm{AP}$. $\therefore 2 \mathrm{~B}=\mathrm{A}+\mathrm{C}$ on adding $\mathrm{B}$ both the sides,
<br/> <br/>we get
<br/> <br/>$$
<br/> <br/>\begin{array}{l}
<br/> <br/>3 \mathrm{~B}=\mathrm{A}+\mathrm{B}+\mathrm{C} \\
<br/> <br/>\Rightarrow 3 \mathrm{~B}=180^{\circ} \Rightarrow \mathrm{B}=60^{\circ}
<br/> <br/>\end{array}
<br/> <br/>$$
<br/> <br/>Now, we know $\cos \mathrm{B}=\frac{\mathrm{a}^{2}+\mathrm{c}^{2}-\mathrm{b}^{2}}{2 \mathrm{ac}}$
<br/> <br/>$$
<br/> <br/>\begin{array}{l}
<br/> <br/>\Rightarrow \cos 60^{\circ}=\frac{10^{2}+\mathrm{c}^{2}-9^{2}}{2 \times 10 \times \mathrm{c}} \\
<br/> <br/>\Rightarrow \frac{1}{2}=\frac{100+\mathrm{c}^{2}-81}{20 \mathrm{c}} \\
<br/> <br/>\Rightarrow \mathrm{c}^{2}-10 \mathrm{c}+19=0 \Rightarrow \mathrm{c}=5 \pm \sqrt{6}
<br/> <br/>\end{array}
<br/> <br/>$$</div>


title: in-a-abc-if-a-c-o-s-a-b-c-o-s-b-c-c-o-s-c-and-the-side-a-2-then-area-of-the-triangle-is
slug: in-a-abc-if-a-c-o-s-a-b-c-o-s-b-c-c-o-s-c-and-the-side-a-2-then-area-of-the-triangle-is-61893
tags: ['Mathematics', 'Properties of Triangles', 'BITSAT', 'BITSAT 2016']
source_db: MarksBatch2_P1.db
question:
<div class="question">In a $\triangle \mathrm{ABC},$ if $\frac{\cos A}{\mathrm{a}}=\frac{\cos \mathrm{B}}{\mathrm{b}}=\frac{\cos \mathrm{C}}{\mathrm{c}},$ and the side $\mathrm{a}=2,$ then area of the triangle is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">1</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">2</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\frac{\sqrt{3}}{2}$</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">$\sqrt{3}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\sqrt{3}$</span> </div>

answer:
<div class="solution">$\frac{\cos A}{a}=\frac{\cos B}{b}=\frac{\cos C}{c}$
<br/> <br/>$\Rightarrow \frac{\cos \mathrm{A}}{2 \mathrm{R} \sin \mathrm{A}}=\frac{\cos \mathrm{B}}{2 \mathrm{R} \sin \mathrm{B}}=\frac{\cos \mathrm{C}}{2 \mathrm{R} \sin \mathrm{C}}$
<br/> <br/>$\Rightarrow \cot A=\cot B=\cot C$
<br/> <br/>$\Rightarrow \mathrm{A}=\mathrm{B}=\mathrm{C}=60^{\circ} \Rightarrow \Delta \mathrm{ABC}$ is equilateral
<br/> <br/>Hence, area of $\Delta=\frac{\sqrt{3}}{4} \mathrm{a}^{2}=\sqrt{3}$.</div>


tags: ['Chemistry', 'Electrochemistry', 'BITSAT', 'BITSAT 2009']
source_db: MarksBatch2_P1.db
title:
in-a-cell-that-utilises-the-reaction-zn-s-2-h-aq-zn-2-aq-h-2-g-addition-of-h-2-so-4-to-cathode-compartment-will-1

slug:
in-a-cell-that-utilises-the-reaction-zn-s-2-h-aq-zn-2-aq-h-2-g-addition-of-h-2-so-4-to-cathode-compartment-will-1-13682

question:
<div class="question">In a cell that utilises the reaction<br/>$\mathrm{Zn}(\mathrm{s})+2 \mathrm{H}^{+}(\mathrm{aq}) \rightarrow \mathrm{Zn}^{2+}(\mathrm{aq})+\mathrm{H}_{2}(\mathrm{~g})$<br/>addition of $\mathrm{H}_{2} \mathrm{SO}_{4}$ to cathode compartment, will</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">increase the $\mathrm{E}$ and shift equilibrium to the right.</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">lower the $E$ and shift equilibrium to the right.</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">lower the $E$ and shift equlibrium to the left.</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">increase the $E$ and shift equilibrium to the left.</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">increase the $\mathrm{E}$ and shift equilibrium to the right.</span> </div>

answer:
<div class="solution">$\mathrm{Zn}(\mathrm{s})+2 \mathrm{H}^{+}+(\mathrm{aq}) \rightleftharpoons$<br/>$$<br/>\mathrm{Zn}^{2+}(\mathrm{aq})+\mathrm{H}_{2}(\mathrm{~g})<br/>$$<br/>$$<br/>\mathrm{E}_{\mathrm{cell}}=\mathrm{E}^{\circ} \mathrm{cell}-\frac{0.059}{2} \log \frac{\left[\mathrm{Zn}^{2+}\right]\left[\mathrm{H}_{2}\right]}{\left[\mathrm{H}^{+}\right]^{2}}<br/>$$<br/>Addition of $\mathrm{H}_{2} \mathrm{SO}_{4}$ will increase $\left[\mathrm{H}^{+}\right]$ and $\mathrm{E}_{\text {cell }}$ will also increase and the equilibrium will shift towards RHS.</div>


tags: ['Chemistry', 'Chemical Equilibrium', 'BITSAT', 'BITSAT 2016']
source_db: MarksBatch2_P1.db
title:
in-a-closed-system-a-s-2-b-g-3-c-g-if-partial-pressure-of-c-is-doubled-then-partial-pressure-of-b-will-be

slug:
in-a-closed-system-a-s-2-b-g-3-c-g-if-partial-pressure-of-c-is-doubled-then-partial-pressure-of-b-will-be-12574

question:
<div class="question">In a closed system, $\mathrm{A}(\mathrm{s}) \rightleftharpoons 2 \mathrm{~B}(\mathrm{~g})+3 \mathrm{C}(\mathrm{g})$, if partial
<br/> <br/>pressure of $\mathrm{C}$ is doubled, then partial pressure of $\mathrm{B}$ will be</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$2 \sqrt{2}$ times the original value</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\frac{1}{2}$ times the original value</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">2 times the original value</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">$\frac{1}{2 \sqrt{2}}$ times the original value</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\frac{1}{2 \sqrt{2}}$ times the original value</span> </div>

answer:
<div class="solution">$\quad K_{p}=p_{B_{1}}^{2} \times p_{C_{1}}^{3}$
<br/> <br/>Again, $K_{p}=p_{B_{2}}^{2} \times\left(2 p_{C_{1}}\right)^{3}$
<br/> <br/>$\therefore \mathrm{p}_{\mathrm{B}_{1}}^{2} \times \mathrm{p}_{\mathrm{C}_{1}}^{3}=\mathrm{P}_{\mathrm{B}_{2}}^{2} \times 8 \mathrm{p}_{\mathrm{C}_{1}}^{3}$
<br/> <br/>$$
<br/> <br/>\therefore \frac{\mathrm{p}_{\mathrm{B}_{1}}^{2}}{8}=\mathrm{p}_{\mathrm{B}_{2}}^{2} \text { or, } \frac{\mathrm{p}_{\mathrm{B}_{1}}}{2 \sqrt{2}}=\mathrm{p}_{\mathrm{B}_{2}}
<br/> <br/>$$</div>


tags: ['Physics', 'Electromagnetic Induction', 'BITSAT', 'BITSAT 2019']
source_db: MarksBatch2_P1.db
title:
in-a-coil-of-resistance-50-the-induced-current-developed-by-changing-magnetic-flux-through-it-is-shown-in-figure-as-a-function-of-time-the-magnitude-o

slug:
in-a-coil-of-resistance-50-the-induced-current-developed-by-changing-magnetic-flux-through-it-is-shown-in-figure-as-a-function-of-time-the-magnitude-o-71140

question:
<div class="question"><p>In a coil of resistance <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>50</mn><mo> </mo><mi mathvariant="normal">Ω</mi></math>, the induced current developed by changing magnetic flux through it, is shown in figure as a function of time. The magnitude of change in flux through the coil is</p><p><img src="https://cdn-question-pool.getmarks.app/pyq/bitsat/8ea035ac-dd7e-479c-8dbe-0b7bb723d4d6-image.png" style="width: 270px; height: 205px;"/></p></div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>10</mn><mo> </mo><mi>Wb</mi></math></span> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>20</mn><mo> </mo><mi>Wb</mi></math></span> </li><li class=""> <span class="option-label">C</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>32</mn><mo> </mo><mi>Wb</mi></math></span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>40</mn><mo> </mo><mi>Wb</mi></math></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>40</mn><mo> </mo><mi>Wb</mi></math></span> </div>

answer:
<div class="solution"><p>According to the given graph <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mi>I</mi><mo>-</mo><mi>t</mi></mrow></mfenced></math> induced current <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mi>I</mi><mo>)</mo></math> is obtained from the slope. Hence, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>I</mi><mo>=</mo><mfrac><mn>1</mn><mi>R</mi></mfrac><mfrac><mrow><mi>d</mi><mi>ϕ</mi></mrow><mrow><mi>d</mi><mi>t</mi></mrow></mfrac></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mi>d</mi><mi>ϕ</mi><mo>=</mo><mi>I</mi><mi>d</mi><mi>t</mi><mo>×</mo><mi>R</mi><mo>=</mo></math> area of triangle <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi><mi>O</mi><mi>S</mi><mo>×</mo><mi>R</mi><mo>=</mo><mfrac><mrow><mn>8</mn><mo>×</mo><mn>0</mn><mo>.</mo><mn>2</mn></mrow><mn>2</mn></mfrac><mo>×</mo><mn>50</mn><mo>=</mo><mn>40</mn><mo> </mo><mi>Wb</mi></math></p></div>


tags: ['Chemistry', 'Solid State', 'BITSAT', 'BITSAT 2019']
source_db: MarksBatch2_P1.db
title:
in-a-compound-atoms-of-elements-y-form-ccp-lattice-and-those-of-element-x-occupy-2-3-rd-of-tetrahedral-voids-the-formula-of-the-compound-can-be

slug:
in-a-compound-atoms-of-elements-y-form-ccp-lattice-and-those-of-element-x-occupy-2-3-rd-of-tetrahedral-voids-the-formula-of-the-compound-can-be-67456

question:
<div class="question">In a compound, atoms of elements <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">Y</mi></math> form ccp lattice and those of element <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">X</mi></math> occupy <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><mo>/</mo><mn>3</mn><mo> </mo><mi>rd</mi></math> of tetrahedral voids. The formula of the compound can be</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi mathvariant="normal">X</mi><mn>4</mn></msub><msub><mi mathvariant="normal">Y</mi><mn>3</mn></msub></math></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi mathvariant="normal">X</mi><mn>2</mn></msub><msub><mi mathvariant="normal">Y</mi><mn>3</mn></msub></math></span> </li><li class=""> <span class="option-label">C</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi mathvariant="normal">X</mi><mn>2</mn></msub><mi mathvariant="normal">Y</mi></math></span> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi mathvariant="normal">X</mi><mn>3</mn></msub><msub><mi mathvariant="normal">Y</mi><mn>4</mn></msub></math></span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi mathvariant="normal">X</mi><mn>4</mn></msub><msub><mi mathvariant="normal">Y</mi><mn>3</mn></msub></math></span> </div>

answer:
<div class="solution"><p>Let atoms of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">Y</mi></math> in ccp structure <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>100</mn></math></p><p>Then, number of tetrahedral voids <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>2</mn><mo>×</mo><mn>100</mn></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>∵</mo></math> Number of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">X</mi></math> atoms <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfrac><mn>2</mn><mn>3</mn></mfrac></math> rd of tetrahedral voids</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>∴</mo></math> Number of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">X</mi></math> atoms <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfrac><mn>2</mn><mn>3</mn></mfrac><mo>×</mo><mn>200</mn><mo>=</mo><mfrac><mn>400</mn><mn>3</mn></mfrac></math></p><p>Hence, formula of compound can be <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><msub><mi mathvariant="normal">X</mi><mn>4</mn></msub><msub><mi mathvariant="normal">Y</mi><mn>3</mn></msub></math>.</p></div>


tags: ['Chemistry', 'Some Basic Concepts of Chemistry', 'BITSAT', 'BITSAT 2019']
source_db: MarksBatch2_P1.db
title:
in-a-compound-c-h-and-n-are-present-in-9-1-3-5-by-weight-if-molecular-weight-of-compound-is-108-the-molecular-formula-of-compound-is

slug:
in-a-compound-c-h-and-n-are-present-in-9-1-3-5-by-weight-if-molecular-weight-of-compound-is-108-the-molecular-formula-of-compound-is-55974

question:
<div class="question">In a compound <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">C</mi><mo>,</mo><mo> </mo><mi mathvariant="normal">H</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">N</mi></math> are present in <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>9</mn><mo>:</mo><mn>1</mn><mo>:</mo><mn>3</mn><mo>.</mo><mn>5</mn></math> by weight. If molecular weight of compound is <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>108</mn></math>, the molecular formula of compound is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi mathvariant="normal">C</mi><mn>2</mn></msub><msub><mi mathvariant="normal">H</mi><mn>6</mn></msub><mo> </mo><msub><mi mathvariant="normal">N</mi><mn>2</mn></msub></math></span> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi mathvariant="normal">C</mi><mn>3</mn></msub><msub><mi mathvariant="normal">H</mi><mn>4</mn></msub><mo> </mo><mi mathvariant="normal">N</mi></math></span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi mathvariant="normal">C</mi><mn>6</mn></msub><msub><mi mathvariant="normal">H</mi><mn>8</mn></msub><mo> </mo><msub><mi mathvariant="normal">N</mi><mn>2</mn></msub></math></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi mathvariant="normal">C</mi><mn>9</mn></msub><msub><mi mathvariant="normal">H</mi><mn>12</mn></msub><mo> </mo><msub><mi mathvariant="normal">N</mi><mn>3</mn></msub></math></span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi mathvariant="normal">C</mi><mn>6</mn></msub><msub><mi mathvariant="normal">H</mi><mn>8</mn></msub><mo> </mo><msub><mi mathvariant="normal">N</mi><mn>2</mn></msub></math></span> </div>

answer:
<div class="solution"><table border="1" cellpadding="1" cellspacing="1" style="width: 500px;"> <tbody> <tr> <td>Element</td> <td>Ratio (by weight)</td> <td>Atomic weight</td> <td>Mol<br/>			Ratio</td> <td>Simplest Mol Ratio</td> </tr> <tr> <td><math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">C</mi></math></td> <td><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>9</mn></math></td> <td><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>12</mn></math></td> <td><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>9</mn><mn>12</mn></mfrac><mo>=</mo><mn>0</mn><mo>.</mo><mn>75</mn></math></td> <td><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mn>0</mn><mo>.</mo><mn>75</mn></mrow><mrow><mn>0</mn><mo>.</mo><mn>25</mn></mrow></mfrac><mo>=</mo><mn>3</mn></math></td> </tr> <tr> <td><math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">H</mi></math></td> <td><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn></math></td> <td><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn></math></td> <td><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>1</mn><mn>1</mn></mfrac><mo>=</mo><mn>1</mn><mo>.</mo><mn>00</mn></math></td> <td><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mn>1</mn><mo>.</mo><mn>00</mn></mrow><mrow><mn>0</mn><mo>.</mo><mn>25</mn></mrow></mfrac><mo>=</mo><mn>4</mn></math></td> </tr> <tr> <td><math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">N</mi></math></td> <td><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>3</mn><mo>.</mo><mn>5</mn></math></td> <td><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>14</mn></math></td> <td><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mn>3</mn><mo>.</mo><mn>5</mn></mrow><mn>14</mn></mfrac><mo>=</mo><mn>0</mn><mo>.</mo><mn>25</mn></math></td> <td><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mn>0</mn><mo>.</mo><mn>25</mn></mrow><mrow><mn>0</mn><mo>.</mo><mn>25</mn></mrow></mfrac><mo>=</mo><mn>1</mn></math></td> </tr> </tbody></table><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>∴</mo></math> Empirical formula <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><msub><mi mathvariant="normal">C</mi><mn>3</mn></msub><msub><mi mathvariant="normal">H</mi><mn>4</mn></msub><mo> </mo><mi mathvariant="normal">N</mi></math></p><p>The empirical formula mass <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>3</mn><mo>×</mo><mn>12</mn><mo>+</mo><mn>4</mn><mo>×</mo><mn>1</mn><mo>+</mo><mn>14</mn><mo>=</mo><mn>54</mn><mo> </mo><mi mathvariant="normal">g</mi></math></p><p>Given, molecular mass <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>108</mn><mo> </mo><mi mathvariant="normal">g</mi></math></p><p>Thus, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">n</mi><mo>=</mo><mfrac><mn>108</mn><mn>54</mn></mfrac><mo>=</mo><mn>2</mn></math></p><p>Hence, molecular formula = (Empirical formula) <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>×</mo><mn>2</mn></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><msub><mi mathvariant="normal">C</mi><mn>3</mn></msub><msub><mi mathvariant="normal">H</mi><mn>4</mn></msub><mo> </mo><mi mathvariant="normal">N</mi></mrow></mfenced><mo>×</mo><mn>2</mn><mo>=</mo><msub><mi mathvariant="normal">C</mi><mn>6</mn></msub><msub><mi mathvariant="normal">H</mi><mn>8</mn></msub><mo> </mo><msub><mi mathvariant="normal">N</mi><mn>2</mn></msub></math></p></div>


tags: ['Physics', 'Electrostatics', 'BITSAT', 'BITSAT 2023 (Memory Based Paper 2)']
source_db: MarksBatch2_P1.db
title:
in-a-cuboid-of-dimension-2-l-2-l-l-a-charge-q-is-placed-at-the-centre-of-the-surface-s-having-area-of-4-l-2-the-flux-through-the-opposite-surface-to-s-1

slug:
in-a-cuboid-of-dimension-2-l-2-l-l-a-charge-q-is-placed-at-the-centre-of-the-surface-s-having-area-of-4-l-2-the-flux-through-the-opposite-surface-to-s-1-33995

question:
<div class="question">In a cuboid of dimension $2 \mathrm{~L} \times 2 \mathrm{~L} \times \mathrm{L}$, a charge $q$ is placed at the centre of the surface ' $S$ ' having area of $4 \mathrm{~L}^2$. The flux through the opposite surface to ' $\mathrm{S}$ ' is given by</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\frac{\mathrm{q}}{12 \varepsilon_0}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\frac{\mathrm{q}}{3 \varepsilon_0}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\frac{\mathrm{q}}{2 \varepsilon_0}$</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">$\frac{q}{6 \varepsilon_0}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\frac{q}{6 \varepsilon_0}$</span> </div>

answer:
<div class="solution">After placing similar cubic at the bottom, we get cube of side $2 \mathrm{~L}$ having change $\mathrm{q}$ at its centre.<br/><br/>From the Gauss's law electric flux through whole<br/><img src="https://cdn-question-pool.getmarks.app/pyq/bitsat/SzzTvHFauXWrbQsPicAom_uB5-dLuGGyZuCfDeMd-ys.original.fullsize.png"/><br/><br/>Flux passing through shaded face<br/>$\phi=\frac{\mathrm{q} / \varepsilon_0}{6}=\frac{\mathrm{q}}{6 \varepsilon_0}$</div>


tags: ['Physics', 'Electrostatics', 'JEE Main']
source_db: MarksBatch2_P1.db
title:
in-a-cuboid-of-dimension-2-l-2-l-l-a-charge-q-is-placed-at-the-centre-of-the-surface-s-having-area-of-4-l-2-the-flux-through-the-opposite-surface-to-s-2

slug:
in-a-cuboid-of-dimension-2-l-2-l-l-a-charge-q-is-placed-at-the-centre-of-the-surface-s-having-area-of-4-l-2-the-flux-through-the-opposite-surface-to-s-2-57329

question:
<div class="question">In a cuboid of dimension $2 \mathrm{~L} \times 2 \mathrm{~L} \times \mathrm{L}$, a charge $q$ is placed at the centre of the surface ' $S$ ' having area of $4 \mathrm{~L}^2$. The flux through the opposite surface to ' $\mathrm{S}$ ' is given by</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\frac{\mathrm{q}}{12 \varepsilon_0}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\frac{\mathrm{q}}{3 \varepsilon_0}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\frac{\mathrm{q}}{2 \varepsilon_0}$</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">$\frac{q}{6 \varepsilon_0}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\frac{q}{6 \varepsilon_0}$</span> </div>

answer:
<div class="solution">After placing similar cubic at the bottom, we get cube of side $2 \mathrm{~L}$ having change $\mathrm{q}$ at its centre.<br/><br/>From the Gauss's law electric flux through whole<br/><img src="https://cdn-question-pool.getmarks.app/pyq/bitsat/SzzTvHFauXWrbQsPicAom_uB5-dLuGGyZuCfDeMd-ys.original.fullsize.png"/><br/><br/>Flux passing through shaded face<br/>$\phi=\frac{\mathrm{q} / \varepsilon_0}{6}=\frac{\mathrm{q}}{6 \varepsilon_0}$</div>


title: in-a-first-order-reaction-with-time-the-concentration-of-the-reactant-decreases
slug: in-a-first-order-reaction-with-time-the-concentration-of-the-reactant-decreases-34699
tags: ['Chemistry', 'Chemical Kinetics', 'BITSAT', 'BITSAT 2015']
source_db: MarksBatch2_P1.db
question:
<div class="question">In a first order reaction with time the concentration of the reactant decreases</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">linearly</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">exponentially</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">no change</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">None of these</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">exponentially</span> </div>

answer:
<div class="solution">For first order reaction $[\mathrm{A}]=\left[\mathrm{A}_{0}\right] \mathrm{e}^{\mathrm{k}}$
<br/> <br/>$\therefore$ The concentration of reactants will exponentially decreases with time.</div>


title: in-a-given-network-the-equivalent-capacitance-between-a-and-b-is-c-1-c-4-1-f-c-2-c-3-2-f
slug: in-a-given-network-the-equivalent-capacitance-between-a-and-b-is-c-1-c-4-1-f-c-2-c-3-2-f-76058
tags: ['Physics', 'Capacitance', 'BITSAT', 'BITSAT 2021']
source_db: MarksBatch2_P1.db
question:
<div class="question">In a given network the equivalent capacitance between $A$ and $B$ is $C_{1}=C_{4}=1 \mu F, C_{2}=C_{3}=2 \mu F$.
<br/> <br/><img src="https://cdn-question-pool.getmarks.app/pyq/bitsat/UR7fH5N3eUVaBtywXMK5A3XxXEaBTe-YDzT_Imfa3NA.original.fullsize.png"/></div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$3 \mu \mathrm{F}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$6 \mu \mathrm{F}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$4.5 \mu \mathrm{F}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$2.5 \mu \mathrm{F}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$3 \mu \mathrm{F}$</span> </div>

answer:
<div class="solution">The equivalent circuit is shown in figure.
<br/> <br/><img src="https://cdn-question-pool.getmarks.app/pyq/bitsat/JJ0PmMzYakzoR5r60KIBHbJN9RsEesxnbhvs4TT-7M0.original.fullsize.png"/><br/> <br/> <br/>$\mathrm{C}_{\mathrm{AB}}=1+\frac{2 \times 2}{2+2}+1$
<br/> <br/>$\mathrm{C}_{\mathrm{AB}}=3 \mu \mathrm{F}$</div>


title: in-a-homonuclear-molecule-which-of-the-following-set-of-orbitals-is-degenerated
slug: in-a-homonuclear-molecule-which-of-the-following-set-of-orbitals-is-degenerated-92082
tags: ['Chemistry', 'Structure of Atom', 'BITSAT', 'BITSAT 2019']
source_db: MarksBatch2_P1.db
question:
<div class="question">In a homonuclear molecule, which of the following set of orbitals is degenerated?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">σ</mi><mn>1</mn><mi mathvariant="normal">s</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">σ</mi><mn>2</mn><mi mathvariant="normal">s</mi></math></span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">π</mi><mn>2</mn><msub><mi mathvariant="normal">p</mi><mi mathvariant="normal">x</mi></msub></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">π</mi><mn>2</mn><msub><mi mathvariant="normal">p</mi><mi mathvariant="normal">y</mi></msub></math></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi mathvariant="normal">π</mi><mo>*</mo></msup><mn>2</mn><msub><mi mathvariant="normal">p</mi><mi mathvariant="normal">x</mi></msub></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi mathvariant="normal">σ</mi><mo>*</mo></msup><mn>2</mn><msub><mi mathvariant="normal">p</mi><mi mathvariant="normal">z</mi></msub></math></span> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">σ</mi><mn>2</mn><msub><mi mathvariant="normal">p</mi><mi mathvariant="normal">x</mi></msub></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi mathvariant="normal">σ</mi><mo>*</mo></msup><mn>2</mn><msub><mi mathvariant="normal">p</mi><mi mathvariant="normal">z</mi></msub></math></span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">π</mi><mn>2</mn><msub><mi mathvariant="normal">p</mi><mi mathvariant="normal">x</mi></msub></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">π</mi><mn>2</mn><msub><mi mathvariant="normal">p</mi><mi mathvariant="normal">y</mi></msub></math></span> </div>

answer:
<div class="solution"><blockquote><p>Orbitals of equal energy are called degenerated orbitals. Homonuclear molecules are the molecules composed of only one type of element and contain following set of degenerated orbitals:</p><p>As, energy of,</p><p> <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">σ</mi><mn>2</mn><mi mathvariant="normal">s</mi><mo>&gt;</mo><mi mathvariant="normal">σ</mi><mn>1</mn><mi mathvariant="normal">s</mi></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">π</mi><mn>2</mn><msub><mi mathvariant="normal">p</mi><mi mathvariant="normal">x</mi></msub><mo>≈</mo><mn>2</mn><msub><mi mathvariant="normal">p</mi><mi mathvariant="normal">y</mi></msub></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi mathvariant="normal">π</mi><mo>*</mo></msup><mn>2</mn><msub><mi mathvariant="normal">p</mi><mi mathvariant="normal">x</mi></msub><mo>&lt;</mo><msup><mi mathvariant="normal">σ</mi><mo>*</mo></msup><mn>2</mn><msub><mi mathvariant="normal">p</mi><mi mathvariant="normal">z</mi></msub></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">σ</mi><mn>2</mn><msub><mi mathvariant="normal">p</mi><mi mathvariant="normal">x</mi></msub><mo>&lt;</mo><msup><mi mathvariant="normal">σ</mi><mo>*</mo></msup><mn>2</mn><msub><mi mathvariant="normal">p</mi><mi mathvariant="normal">z</mi></msub></math></p><p>Thus, only <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">π</mi><mn>2</mn><msub><mi mathvariant="normal">p</mi><mi mathvariant="normal">x</mi></msub></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">π</mi><mn>2</mn><msub><mi mathvariant="normal">p</mi><mi mathvariant="normal">y</mi></msub></math> are degenerate orbitals.</p></blockquote></div>


title: in-a-monoclinic-unit-cell-the-relation-of-sides-and-angles-are-respectively-1
slug: in-a-monoclinic-unit-cell-the-relation-of-sides-and-angles-are-respectively-1-75819
tags: ['Chemistry', 'Solid State', 'BITSAT', 'BITSAT 2020']
answer: <div class="solution">No Solution</div>
source_db: MarksBatch2_P1.db
question:
<div class="question">In a monoclinic unit cell, the relation of sides and angles are respectively:</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$a=b \neq c$ and $\alpha=\beta=\gamma=90^{\circ}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$a \neq b \neq c$ and $\alpha=\beta=\gamma=90^{\circ}$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$a \neq b \neq c$ and $\beta=\gamma=90^{\circ} \neq \alpha$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$a \neq b \neq c$ and $\alpha \neq \beta \neq \gamma \neq 90^{\circ}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$a \neq b \neq c$ and $\beta=\gamma=90^{\circ} \neq \alpha$</span> </div>


tags: ['Physics', 'Dual Nature of Matter', 'BITSAT', 'BITSAT 2009']
source_db: MarksBatch2_P1.db
title:
in-a-photoelectric-experiment-with-light-of-wavelength-the-fastest-electron-has-speed-v-if-the-exciting-wavelength-is-changed-to-3-4-the-speed-of-the-

slug:
in-a-photoelectric-experiment-with-light-of-wavelength-the-fastest-electron-has-speed-v-if-the-exciting-wavelength-is-changed-to-3-4-the-speed-of-the-56577

question:
<div class="question">In a photoelectric experiment, with light of wavelength $\lambda,$ the fastest electron has speed $v$. If the exciting wavelength is changed to $3 \lambda / 4$, the speed of the fastest emitted electron will become</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$v \sqrt{\frac{3}{4}}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$v \sqrt{\frac{4}{3}}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">less than $v \sqrt{\frac{4}{3}}$</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">greater than $v \sqrt{\frac{4}{3}}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">greater than $v \sqrt{\frac{4}{3}}$</span> </div>

answer:
<div class="solution">$\frac{1}{2} \mathrm{mv}^{2}=\frac{\mathrm{hc}}{\lambda}-\phi$<br/>$$<br/>\begin{array}{l}<br/>\frac{1}{2} \mathrm{~m}^{\prime}=\frac{\mathrm{hc}}{(3 \lambda / 4)}-\phi=\frac{4 \mathrm{hc}}{3 \lambda}-\phi \\<br/>\text { Clearly, } \mathrm{v}^{\prime}&gt;\sqrt{\frac{4}{3} \mathrm{v}}<br/>\end{array}<br/>$$</div>


tags: ['Mathematics', 'Permutation Combination', 'JEE Main']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">20</span> </div>
source_db: MarksBatch2_P1.db
title:
in-a-polygon-no-three-diagonals-are-concurrent-if-the-total-number-of-points-of-intersection-of-diagonals-interior-to-the-polygon-be-70-then-the-numbe-1

slug:
in-a-polygon-no-three-diagonals-are-concurrent-if-the-total-number-of-points-of-intersection-of-diagonals-interior-to-the-polygon-be-70-then-the-numbe-1-91920

question:
<div class="question">In a polygon no three diagonals are concurrent. If the total number of points of intersection of diagonals interior to the polygon be 70 then the number of diagonals of the polygon is</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">20</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">28</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">8</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">None of these</span> </li> </ul>

answer:
<div class="solution">A combination of four vertices is equivalent to one interior point of intersection of diagonals.<br/><img src="https://cdn-question-pool.getmarks.app/pyq/bitsat/rPkMaoMA84X7jjk_MZSrQoolKg3Y1oDMudbyYmES1nc.original.fullsize.png"/><br/><br/>\(\therefore\) No. of interior points of intersection<br/>\(\begin{aligned}<br/>&amp; ={ }^{\mathrm{n}} \mathrm{C}_4=70 \\<br/>&amp; \Rightarrow \mathrm{n}(\mathrm{n}-1)(\mathrm{n}-2)(\mathrm{n}-3)=5.6 .7 .8 \\<br/>&amp; \therefore \mathrm{n}=8<br/>\end{aligned}\)<br/>So, number of diagonals \(={ }^8 \mathrm{C}_2-8=20\)</div>


tags: ['Mathematics', 'Permutation Combination', 'BITSAT', 'BITSAT 2010']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">20</span> </div>
source_db: MarksBatch2_P1.db
title:
in-a-polygon-no-three-diagonals-are-concurrent-if-the-total-number-of-points-of-intersection-of-diagonals-interior-to-the-polygon-be-70-then-the-numbe

slug:
in-a-polygon-no-three-diagonals-are-concurrent-if-the-total-number-of-points-of-intersection-of-diagonals-interior-to-the-polygon-be-70-then-the-numbe-29615

question:
<div class="question">In a polygon no three diagonals are concurrent. If the total number of points of intersection of diagonals interior to the polygon be 70 then the number of diagonals of the polygon is</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">20</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">28</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">8</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">None of these</span> </li> </ul>

answer:
<div class="solution">A combination of four vertices is equivalent to one interior point of intersection of diagonals.<br/><img src="https://cdn-question-pool.getmarks.app/pyq/bitsat/rPkMaoMA84X7jjk_MZSrQoolKg3Y1oDMudbyYmES1nc.original.fullsize.png"/><br/><br/>\(\therefore\) No. of interior points of intersection<br/>\(\begin{aligned}<br/>&amp; ={ }^{\mathrm{n}} \mathrm{C}_4=70 \\<br/>&amp; \Rightarrow \mathrm{n}(\mathrm{n}-1)(\mathrm{n}-2)(\mathrm{n}-3)=5.6 .7 .8 \\<br/>&amp; \therefore \mathrm{n}=8<br/>\end{aligned}\)<br/>So, number of diagonals \(={ }^8 \mathrm{C}_2-8=20\)</div>


tags: ['Physics', 'Nuclear Physics', 'BITSAT', 'BITSAT 2022']
source_db: MarksBatch2_P1.db
title:
in-a-radioactive-material-the-activity-at-time-t-1-is-a-1-and-at-a-later-time-t-2-it-is-a-2-if-the-decay-constant-of-the-material-is-then

slug:
in-a-radioactive-material-the-activity-at-time-t-1-is-a-1-and-at-a-later-time-t-2-it-is-a-2-if-the-decay-constant-of-the-material-is-then-56046

question:
<div class="question">In a radioactive material the activity at time $t_1$, is $A_1$ and at a later time $t_2$, it is $A_2$. If the decay constant of the material is $\lambda$, then</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data"> <br/>$A_1=A_2 e^{-\lambda\left(t_1-t_2\right)}$
<br/></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data"> <br/>$A_1=A_2 e^{\lambda\left(t_1-t_2\right)}$
<br/></span> </li><li class=""> <span class="option-label">C</span> <span class="option-data"> <br/>$A_1=A_2\left(t_2 / t_1\right)$
<br/></span> </li><li class=""> <span class="option-label">D</span> <span class="option-data"> <br/>$A_1=A_2$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"> <br/>$A_1=A_2 e^{-\lambda\left(t_1-t_2\right)}$
<br/></span> </div>

answer:
<div class="solution">From radioactive decay law,
<br/>$$
<br/>-\frac{d N}{d t} \propto N \text { or }-\frac{d N}{d t}=\lambda N
<br/>$$
<br/>Thus,
<br/>$$
<br/>A=-\frac{d N}{d t} \text { or } A=\lambda N
<br/>$$
<br/>$$
<br/>\Rightarrow \quad A=\lambda N_0 e^{-\lambda t}
<br/>$$
<br/>Where, $A_0=\lambda N_0$ is the activity of the radioactive material at time, $t=0$
<br/>At time, $t_1 \quad A_1=A_0 e^{-\lambda t_1}$
<br/>At time, $t_2 \quad A_2=A_0 e^{-\lambda t_2}$
<br/>On dividing Eq. (ii) by Eq. (iii), we have
<br/>$$
<br/>\begin{aligned}
<br/>\quad \frac{A_1}{A_2} &amp; =\frac{e^{-\lambda t_1}}{e^{-\lambda t_2}}=e^{-\lambda\left(t_1-t_2\right)} \\
<br/>\Rightarrow \quad \quad A_1 &amp; =A_2 e^{-\lambda\left(t_1-t_2\right)}
<br/>\end{aligned}
<br/>$$</div>


title: in-a-reflecting-telescope-a-secondary-mirror-is-used-to-1
slug: in-a-reflecting-telescope-a-secondary-mirror-is-used-to-1-44659
question: <div class="question">In a reflecting telescope, a secondary mirror is used to:</div>
tags: ['Physics', 'Ray Optics', 'BITSAT', 'BITSAT 2023 (Memory Based Paper 1)']
source_db: MarksBatch2_P1.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">reduce the problem of mechanical support</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">remove spherical aberration</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">make chromatic aberration zero</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">move the eyepiece outside the telescopic tube</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">move the eyepiece outside the telescopic tube</span> </div>

answer:
<div class="solution">(d) To redirect the light that enters the telescope to the eyepiece or camera. The primary mirror of a reflecting telescopes gathers the light and reflects towards the secondary mirror which then reflect the light towards the eyepiece allowing the observer to see image.<br/>It has advantage of a large focal length in a short telescope.</div>


tags: ['Physics', 'Electrostatics', 'BITSAT', 'BITSAT 2012']
source_db: MarksBatch2_P1.db
title:
in-a-region-of-space-having-a-uniform-electric-field-e-a-hemispherical-bowl-of-radius-r-is-placed-the-electric-flux-through-the-bowl-is

slug:
in-a-region-of-space-having-a-uniform-electric-field-e-a-hemispherical-bowl-of-radius-r-is-placed-the-electric-flux-through-the-bowl-is-42259

question:
<div class="question">In a region of space having a uniform electric field $\mathrm{E}$, a hemispherical bowl of radius $\mathrm{r}$ is placed. The electric flux $\phi$ through the bowl is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$2 \pi \mathrm{RE}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$4 \pi R^{2} E$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$2 \pi R^{2} E$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\pi R^{2} E$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$2 \pi R^{2} E$</span> </div>

answer:
<div class="solution">$\phi=\mathrm{E}(\mathrm{ds}) \cos \theta=\mathrm{E}\left(2 \pi \mathrm{r}^{2}\right) \cos 0^{\circ}=2 \pi \mathrm{r}^{2} \mathrm{E}$</div>


tags: ['Physics', 'Electrostatics', 'BITSAT', 'BITSAT 2010']
source_db: MarksBatch2_P1.db
title:
in-a-region-the-potential-is-represented-by-v-x-y-z-6-x-8-xy-8-y-6-yz-where-v-is-in-volts-and-x-y-z-are-in-metres-the-electric-force-experienced-by-a-

slug:
in-a-region-the-potential-is-represented-by-v-x-y-z-6-x-8-xy-8-y-6-yz-where-v-is-in-volts-and-x-y-z-are-in-metres-the-electric-force-experienced-by-a-96368

question:
<div class="question">In a region, the potential is represented by $\mathrm{V}(\mathrm{x}, \mathrm{y}, \mathrm{z})=6 \mathrm{x}-8 \mathrm{xy}-8 \mathrm{y}+6 \mathrm{yz},$ where $\mathrm{V}$ is in volts and $\mathrm{x}, \mathrm{y}, \mathrm{z}$ are in metres. The electric force experienced by a charge of 2 coulomb situated at point (1,1,1) is :</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$6 \sqrt{5} \mathrm{~N}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$30 \mathrm{~N}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$24 \mathrm{~N}$</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">$4 \sqrt{35} \mathrm{~N}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$4 \sqrt{35} \mathrm{~N}$</span> </div>

answer:
<div class="solution">$\overrightarrow{\mathrm{E}}=-\frac{\partial \mathrm{V}}{\partial \mathrm{x}} \hat{\mathrm{i}}-\frac{\partial \mathrm{V}}{\partial \mathrm{y}} \hat{\mathrm{j}}-\frac{\partial \mathrm{V}}{\partial \mathrm{z}} \hat{\mathrm{k}}$<br/>$=-[(6-8 \mathrm{y}) \hat{\mathrm{i}}+(-8 \mathrm{x}-8+6 \mathrm{z}) \hat{\mathrm{j}}+(6 \mathrm{y}) \hat{\mathrm{k}}]$<br/>$\mathrm{At}(1,1,1), \overrightarrow{\mathrm{E}}=2 \hat{\mathrm{i}}+10 \hat{\mathrm{j}}-6 \hat{\mathrm{k}}$<br/>$\Rightarrow \quad(\overrightarrow{\mathrm{E}})=\sqrt{2^{2}+10^{2}+6^{2}}=\sqrt{140}=2 \sqrt{35}$<br/>$\therefore \quad \mathrm{F}=\mathrm{q} \overrightarrow{\mathrm{E}}=2 \times 2 \sqrt{35}=4 \sqrt{35}$</div>


tags: ['Physics', 'Oscillations', 'BITSAT', 'BITSAT 2022']
source_db: MarksBatch2_P1.db
title:
in-a-resonance-column-first-and-second-resonance-are-obtained-at-depths-24-cm-and-78-cm-the-third-resonance-will-be-obtained-at-depth

slug:
in-a-resonance-column-first-and-second-resonance-are-obtained-at-depths-24-cm-and-78-cm-the-third-resonance-will-be-obtained-at-depth-83975

question:
<div class="question">In a resonance column first and second resonance are obtained at depths $24 \mathrm{~cm}$ and $78 \mathrm{~cm}$ the third resonance will be obtained at depth.</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"> <br/>$160 \mathrm{~cm}$
<br/></span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data"> <br/>$132 \mathrm{~cm}$
<br/></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data"> <br/>$131 \mathrm{~cm}$
<br/></span> </li><li class=""> <span class="option-label">D</span> <span class="option-data"> <br/>$152 \mathrm{~cm}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"> <br/>$132 \mathrm{~cm}$
<br/></span> </div>

answer:
<div class="solution">Given, first resonance column at depth, $l_1=24 \mathrm{~cm}$
<br/>Second resonance column at depth, $l_2=78 \mathrm{~cm}$
<br/>Third resonance column at depths, $l_3=$ ?
<br/>As
<br/>$$
<br/>\begin{aligned}
<br/>&amp; l_1+x=\frac{\lambda}{4}=24 \quad \quad \ldots (i)\\
<br/>&amp; l_2+x=\frac{3 \lambda}{4}=78 \quad \quad \ldots (ii)\\
<br/>&amp; l_3+x=\frac{5 \lambda}{4}  \quad \quad \ldots (iii)
<br/>\end{aligned}
<br/>$$
<br/>From Eqs. (i) and (ii),
<br/>$$
<br/>x=\frac{l_2-3 l_1}{2}=\frac{78-3(24)}{2}=3 \quad \quad \ldots (iv)
<br/>$$
<br/>From Eqs. (iii) and (i), $\frac{l_3+x}{l_1+x}=5$
<br/>$$
<br/>l_3=5 l_1+5 x-x=5 l_1+4 x
<br/>$$
<br/>Substituting $l_1$ and $x$ value, we get
<br/>$$
<br/>l_3=5(24)+4(3) \Rightarrow l_3=132 \mathrm{~cm} .
<br/>$$</div>


tags: ['Chemistry', 'Haloalkanes and Haloarenes', 'JEE Main']
source_db: MarksBatch2_P1.db
title:
in-a-s-n-2-substitution-reaction-of-the-type-r-br-cl-dmf-r-cl-br-which-one-of-the-following-has-the-highest-relative-rate-2

slug:
in-a-s-n-2-substitution-reaction-of-the-type-r-br-cl-dmf-r-cl-br-which-one-of-the-following-has-the-highest-relative-rate-2-98343

question:
<div class="question">In a $\mathrm{S}_{\mathrm{N}} 2$ substitution reaction of the type $\mathrm{R}-\mathrm{Br}+\mathrm{Cl}^{-} \stackrel{\mathrm{DMF}}{\longrightarrow} \mathrm{R}-\mathrm{Cl}+\mathrm{Br}^{-}$
<br/> <br/>which one of the following has the highest relative rate?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\mathrm{CH}_{3}-\mathrm{CH}_{2}-\mathrm{CH}_{2} \mathrm{Br}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><img src="https://cdn-question-pool.getmarks.app/pyq/bitsat/d1XsGBHA2CzZm0OetNdOJst_dlnO_s2lcVzKm3XRcQI.original.fullsize.png"/></span> </li><li class=""> <span class="option-label">C</span> <span class="option-data"><img src="https://cdn-question-pool.getmarks.app/pyq/bitsat/SRpALsUieGgskOCrw7D3Le2-4ZWmZcF-ctXhDUrKOOQ.original.fullsize.png"/></span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">$\mathrm{CH}_{3} \mathrm{CH}_{2} \mathrm{Br}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\mathrm{CH}_{3} \mathrm{CH}_{2} \mathrm{Br}$</span> </div>

answer:
<div class="solution">$\mathrm{S}_{\mathrm{N}} 2$ mechanism is followed in case of primary and secondary halides i.e., $\mathrm{S}_{\mathrm{N}}^{2}$ reaction is favoured by small groups on the carbon atom attached to halogens so
<br/> <br/><img src="https://cdn-question-pool.getmarks.app/pyq/bitsat/UxGpTB0zlK2tNKPe8q7QtZVHOXwWyjcPieTaY2aLHc4.original.fullsize.png"/></div>


tags: ['Chemistry', 'Haloalkanes and Haloarenes', 'BITSAT', 'BITSAT 2013']
source_db: MarksBatch2_P1.db
title:
in-a-s-n-2-substitution-reaction-of-the-type-r-br-cl-dmf-r-cl-br-which-one-of-the-following-has-the-highest-relative-rate

slug:
in-a-s-n-2-substitution-reaction-of-the-type-r-br-cl-dmf-r-cl-br-which-one-of-the-following-has-the-highest-relative-rate-51959

question:
<div class="question">In a $\mathrm{S}_{\mathrm{N}} 2$ substitution reaction of the type $\mathrm{R}-\mathrm{Br}+\mathrm{Cl}^{-} \stackrel{\mathrm{DMF}}{\longrightarrow} \mathrm{R}-\mathrm{Cl}+\mathrm{Br}^{-}$
<br/> <br/>which one of the following has the highest relative rate?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\mathrm{CH}_{3}-\mathrm{CH}_{2}-\mathrm{CH}_{2} \mathrm{Br}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><img src="https://cdn-question-pool.getmarks.app/pyq/bitsat/d1XsGBHA2CzZm0OetNdOJst_dlnO_s2lcVzKm3XRcQI.original.fullsize.png"/></span> </li><li class=""> <span class="option-label">C</span> <span class="option-data"><img src="https://cdn-question-pool.getmarks.app/pyq/bitsat/SRpALsUieGgskOCrw7D3Le2-4ZWmZcF-ctXhDUrKOOQ.original.fullsize.png"/></span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">$\mathrm{CH}_{3} \mathrm{CH}_{2} \mathrm{Br}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\mathrm{CH}_{3} \mathrm{CH}_{2} \mathrm{Br}$</span> </div>

answer:
<div class="solution">$\mathrm{S}_{\mathrm{N}} 2$ mechanism is followed in case of primary and secondary halides i.e., $\mathrm{S}_{\mathrm{N}}^{2}$ reaction is favoured by small groups on the carbon atom attached to halogens so
<br/> <br/><img src="https://cdn-question-pool.getmarks.app/pyq/bitsat/UxGpTB0zlK2tNKPe8q7QtZVHOXwWyjcPieTaY2aLHc4.original.fullsize.png"/></div>


title: in-a-semiconductor-diode-the-barrier-potential-offers-opposition-to
slug: in-a-semiconductor-diode-the-barrier-potential-offers-opposition-to-40543
question: <div class="question">In a semiconductor diode, the barrier potential offers opposition to</div>
tags: ['Physics', 'Semiconductors', 'BITSAT', 'BITSAT 2011']
answer: <div class="solution">No Solution available</div>
source_db: MarksBatch2_P1.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">holes in P-region only</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">free electrons in N-region only</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">majority carriers in both regions</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">majority as well as minority carriers in both regions</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">majority carriers in both regions</span> </div>


tags: ['Mathematics', 'Sequences and Series', 'BITSAT', 'BITSAT 2022']
source_db: MarksBatch2_P1.db
title:
in-a-sequence-of-21-terms-the-first-11-terms-are-in-a-p-with-common-difference-2-and-the-last-11-terms-are-in-gp-with-common-ratio-2-if-the-middle-ter

slug:
in-a-sequence-of-21-terms-the-first-11-terms-are-in-a-p-with-common-difference-2-and-the-last-11-terms-are-in-gp-with-common-ratio-2-if-the-middle-ter-24875

question:
<div class="question">In a sequence of $21$ terms, the first $11$ terms are in $AP$ with common difference $2$ and the last $11$ terms are in $GP$ with common ratio $2$. If the middle term of $AP$ be equal to the middle term of the $GP$, then the middle term of the entire sequence is</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data"> <br/>$-\frac{10}{31}$
<br/></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data"> <br/>$\frac{10}{31}$
<br/></span> </li><li class=""> <span class="option-label">C</span> <span class="option-data"> <br/>$\frac{32}{31}$
<br/></span> </li><li class=""> <span class="option-label">D</span> <span class="option-data"> <br/>$-\frac{31}{32}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"> <br/>$-\frac{10}{31}$
<br/></span> </div>

answer:
<div class="solution">Since, the first $11$ terms are in $AP$, $d=2$
<br/>$\therefore \quad a_{11}=a+10 d=a+20$
<br/>The middle term of $AP$ is
<br/>$T_6=a+5 d=a+10$
<br/>For the next $11$ terms in $GP$
<br/>$r=2$
<br/>$\therefore$ The middle term of $GP$ is $b(2)^5$ where, $b$ is the first term of a $GP$ which is the last term of $AP$
<br/>$b(2)^5=(a+20) 32$
<br/>According to the given condition,
<br/>$\begin{aligned}
<br/>\Rightarrow \quad a+10 &amp; =(a+20) 32 \\
<br/>\Rightarrow \quad 31 a &amp; =10-640 \\
<br/>a &amp; =-\frac{630}{31}
<br/>\end{aligned}$
<br/>$\therefore$ Middle term of entire sequence is $11th$ term
<br/>$\begin{aligned} \therefore \quad T_{11} &amp; =\frac{-630}{31}+10 \times d \\ &amp; =\frac{-630}{31}+10 \times 2=\frac{-10}{31}\end{aligned}$</div>


tags: ['Physics', 'Oscillations', 'BITSAT', 'BITSAT 2021']
source_db: MarksBatch2_P1.db
title:
in-a-simple-pendulum-of-length-i-the-bob-is-pulled-aside-from-its-equilibrium-position-through-an-angle-and-then-released-the-bob-passes-through-the-e

slug:
in-a-simple-pendulum-of-length-i-the-bob-is-pulled-aside-from-its-equilibrium-position-through-an-angle-and-then-released-the-bob-passes-through-the-e-34085

question:
<div class="question">In a simple pendulum of length I the bob is pulled aside from its equilibrium position through an angle $\theta$ and then released. The bob passes through the equilibrium position with speed</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\sqrt{2 g l(1+\cos \theta)}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\sqrt{2 g l \sin \theta}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\sqrt{2 \mathrm{~g} l}$</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">$\sqrt{2 \mathrm{~g} l(1-\cos \theta)}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\sqrt{2 \mathrm{~g} l(1-\cos \theta)}$</span> </div>

answer:
<div class="solution">If $l$ is length of pendulum and $\theta$ be angular amplitude then from principle of conservation of energy
<br/> <br/>(Initial total energy) $0+\mathrm{mgh}=\frac{1}{2} m v^{2}+0$ (Final total energy) $\Rightarrow v=\sqrt{2 g h}=\sqrt{2 g l(1-\cos \theta)}$</div>


tags: ['Chemistry', 'Solid State', 'BITSAT', 'BITSAT 2023 (Memory Based Paper 2)']
source_db: MarksBatch2_P1.db
title:
in-a-solid-lattice-the-cation-has-left-a-lattice-site-and-is-located-at-an-interstitial-position-the-lattice-defect-is

slug:
in-a-solid-lattice-the-cation-has-left-a-lattice-site-and-is-located-at-an-interstitial-position-the-lattice-defect-is-86950

question:
<div class="question">In a solid lattice, the cation has left a lattice site and is located at an interstitial position, the lattice defect is :</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><br/>Interstitial defect</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><br/>Valency defect</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data"><br/>Frenkel defect</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><br/>Schottky defect</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><br/>Frenkel defect</span> </div>

answer:
<div class="solution">Frenkel defect is due to dislocation of ion from its usual lattice site to interstitial position.</div>


tags: ['Mathematics', 'Sets and Relations', 'BITSAT', 'BITSAT 2020']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">41</span> </div>
source_db: MarksBatch2_P1.db
title:
in-a-statistical-investigation-of-1003-families-of-calcutta-it-was-found-that-63-families-has-neither-a-radio-nor-a-tv-794-families-has-a-radio-and-18

slug:
in-a-statistical-investigation-of-1003-families-of-calcutta-it-was-found-that-63-families-has-neither-a-radio-nor-a-tv-794-families-has-a-radio-and-18-21285

question:
<div class="question">In a statistical investigation of 1003 families of Calcutta, it was found that 63 families has neither a radio nor a T.V, 794 families has a radio and 187 has T.V. The number of families in that group having both a radio and a T.V is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">36</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">41</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">32</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">None of these</span> </li> </ul>

answer:
<div class="solution"><img src="https://cdn-question-pool.getmarks.app/pyq/bitsat/9Jni0yQvN0bRVYO4WI8L2yws0JP2EPtvxPPdAPz7vIs.original.fullsize.png"/><br/> <br/> <br/>Let $\mathrm{R}$ be the set of families having a radio and $T$ the set families having a T.V. then $\mathrm{n}(\mathrm{R} \cup \mathrm{T})=$ The number of families having at least on of the radio and $\mathrm{T} \mathrm{V}=1003-63=940$ $n(\mathrm{R})=794$ and $\mathrm{n}(\mathrm{T})=187$
<br/> <br/>Let $x$ families have both a radio and a T.V. Then number of families who have only radio $=$ $794-x$
<br/> <br/>And the number of families who have only T.V. $=187-x$
<br/> <br/>Erom Venn diagram, $794-x+x-187-x=940$ $\Rightarrow 981-x=940$ or $x=981-940=41$
<br/> <br/>Hence, the required number of families having both a radio and a T.V.=41</div>


tags: ['Mathematics', 'Probability', 'BITSAT', 'BITSAT 2018']
source_db: MarksBatch2_P1.db
title:
in-a-test-an-examiner-either-guesses-or-copies-or-knows-the-answer-to-a-multiple-choice-question-with-4-choices-the-probability-that-heshe-makes-a-gue

slug:
in-a-test-an-examiner-either-guesses-or-copies-or-knows-the-answer-to-a-multiple-choice-question-with-4-choices-the-probability-that-heshe-makes-a-gue-69898

question:
<div class="question">In a test an examiner either guesses or copies or knows the answer to a multiple choice question with <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>4</mn></math> choices. The probability that he/she makes a guess is <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>1</mn><mn>3</mn></mfrac></math>. The probability that he/she copies the answer is <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>1</mn><mn>6</mn></mfrac></math>. If the probability that the answer is correct, given that he/she copied it is <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>1</mn><mn>8</mn></mfrac></math>, then the probability that he/she knows the answer to a question given that he/she correctly answered it, is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>27</mn><mn>29</mn></mfrac></math></span> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>26</mn><mn>29</mn></mfrac></math></span> </li><li class=""> <span class="option-label">C</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>25</mn><mn>29</mn></mfrac></math></span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>24</mn><mn>29</mn></mfrac></math></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>24</mn><mn>29</mn></mfrac></math></span> </div>

answer:
<div class="solution"><p>Let <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi></math> be the event that the examinee gives the correct answer.</p><p>Let <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>G</mi><mo>,</mo><mo> </mo><mi>C</mi><mo>,</mo><mo> </mo><mi>K</mi></math> stand for guessing, copying and knowing. respectively.</p><p>Given, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi><mfenced><mi>C</mi></mfenced><mo>=</mo><mfrac><mn>1</mn><mn>6</mn></mfrac><mo>,</mo><mo> </mo><mi>P</mi><mfenced><mi>G</mi></mfenced><mo>=</mo><mfrac><mn>1</mn><mn>3</mn></mfrac></math></p><p>and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi><mfenced><mfrac><mi>A</mi><mi>C</mi></mfrac></mfenced><mo>=</mo><mfrac><mn>1</mn><mn>8</mn></mfrac></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mi>P</mi><mfenced><mi>C</mi></mfenced><mo>+</mo><mi>P</mi><mfenced><mi>G</mi></mfenced><mo>+</mo><mi>P</mi><mfenced><mi>K</mi></mfenced><mo>=</mo><mn>1</mn></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mfrac><mn>1</mn><mn>6</mn></mfrac><mo>+</mo><mfrac><mn>1</mn><mn>3</mn></mfrac><mo>+</mo><mi>P</mi><mfenced><mi>K</mi></mfenced><mo>=</mo><mn>1</mn></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>∴</mo><mo> </mo><mi>P</mi><mfenced><mi>K</mi></mfenced><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></math></p><p>Also, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi><mfenced><mfrac><mi>A</mi><mi>K</mi></mfrac></mfenced><mo>=</mo><mn>1</mn><mo>,</mo></math> for if the examinee knows, he/she will correctly answer it and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi><mfenced><mfrac><mi>A</mi><mi>G</mi></mfrac></mfenced><mo>=</mo><mfrac><mn>1</mn><mn>4</mn></mfrac><mo>,</mo></math></p><p> since there are four choices.</p><p>Now, total probability <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi><mfenced><mi>A</mi></mfenced><mo>=</mo><mi>P</mi><mfenced><mi>G</mi></mfenced><mi>P</mi><mfenced><mfrac><mi>A</mi><mi>G</mi></mfrac></mfenced><mo>+</mo><mi>P</mi><mfenced><mi>C</mi></mfenced><mi>P</mi><mfenced><mfrac><mi>A</mi><mi>C</mi></mfrac></mfenced><mo>+</mo><mi>P</mi><mfenced><mi>K</mi></mfenced><mi>P</mi><mfenced><mfrac><mi>A</mi><mi>K</mi></mfrac></mfenced></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfrac><mn>1</mn><mn>3</mn></mfrac><mo>·</mo><mfrac><mn>1</mn><mn>4</mn></mfrac><mo>+</mo><mfrac><mn>1</mn><mn>6</mn></mfrac><mo>·</mo><mfrac><mn>1</mn><mn>8</mn></mfrac><mo>+</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mo>·</mo><mn>1</mn></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfrac><mn>1</mn><mn>12</mn></mfrac><mo>+</mo><mfrac><mn>1</mn><mn>48</mn></mfrac><mo>+</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mo>=</mo><mfrac><mrow><mn>4</mn><mo>+</mo><mn>1</mn><mo>+</mo><mn>24</mn></mrow><mn>48</mn></mfrac><mo>=</mo><mfrac><mn>29</mn><mn>48</mn></mfrac></math></p><p>Now, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi><mfenced><mfrac><mi>K</mi><mi>A</mi></mfrac></mfenced><mo>=</mo><mfrac><mrow><mi>P</mi><mfenced><mi>K</mi></mfenced><mi>P</mi><mfenced><mfrac><mi>A</mi><mi>K</mi></mfrac></mfenced></mrow><mrow><mi>P</mi><mfenced><mi>A</mi></mfenced></mrow></mfrac></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfrac><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo>·</mo><mn>1</mn></mrow><mfrac><mn>29</mn><mn>48</mn></mfrac></mfrac><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mo>·</mo><mfrac><mn>48</mn><mn>29</mn></mfrac><mo>=</mo><mfrac><mn>24</mn><mn>29</mn></mfrac></math></p></div>


tags: ['Physics', 'Thermodynamics', 'BITSAT', 'BITSAT 2014']
source_db: MarksBatch2_P1.db
title:
in-a-thermodynamic-process-the-pressure-of-a-fixed-mass-of-a-gas-is-changed-in-such-a-manner-that-the-gas-releases-20-j-of-heat-and-8-j-of-work-is-don

slug:
in-a-thermodynamic-process-the-pressure-of-a-fixed-mass-of-a-gas-is-changed-in-such-a-manner-that-the-gas-releases-20-j-of-heat-and-8-j-of-work-is-don-64272

question:
<div class="question">In a thermodynamic process, the pressure of a fixed mass of a gas is changed in such a manner that the gas releases $20 \mathrm{~J}$ of heat and $8 \mathrm{~J}$ of work is done on the gas. If the initial internal energy of the gas was $30 \mathrm{~J}$, then the final internal energy will be</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$2 \mathrm{~J}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$42 \mathrm{~J}$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$18 \mathrm{~J}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$58 \mathrm{~J}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$18 \mathrm{~J}$</span> </div>

answer:
<div class="solution">Given $\Delta \mathrm{Q}=-20 \mathrm{~J}, \mathrm{~W}=-8 \mathrm{~J}$
<br/> <br/>Using Ist law $\Delta \mathrm{Q}=\Delta \mathrm{U}+\Delta \mathrm{W}$
<br/> <br/>$$
<br/> <br/>\begin{array}{l}
<br/> <br/>\Rightarrow \Delta Q=-20+8=-12 \mathrm{~J} \\
<br/> <br/>\mathrm{Uf}=-12+30=18 \mathrm{~J}
<br/> <br/>\end{array}
<br/> <br/>$$</div>


title: in-a-triangle-abc-if-a-2-b-6-0-and-c-7-5-then-b-equals
slug: in-a-triangle-abc-if-a-2-b-6-0-and-c-7-5-then-b-equals-38431
tags: ['Mathematics', 'Properties of Triangles', 'BITSAT', 'BITSAT 2012']
source_db: MarksBatch2_P1.db
question:
<div class="question">In a triangle $\mathrm{ABC},$ if $\mathrm{a}=2, \mathrm{~B}=60^{\circ}$ and $\mathrm{C}=75^{\circ}$, then b equals</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\sqrt{3}$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$\sqrt{6}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\sqrt{9}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$1+\sqrt{2}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\sqrt{6}$</span> </div>

answer:
<div class="solution">$\mathrm{A}=180^{\circ}-60^{\circ}-75^{\circ}=180^{\circ}-135^{\circ}=45^{\circ}$
<br/>Now, $\frac{\mathrm{a}}{\sin \mathrm{A}}=\frac{\mathrm{b}}{\sin \mathrm{B}}$
<br/>$\Rightarrow \frac{2}{\sin 45^{\circ}}=\frac{\mathrm{b}}{\sin 60^{\circ}} \Rightarrow \mathrm{b}=\frac{2 \cdot(\sqrt{3} / 2)}{1 / \sqrt{2}}=\sqrt{6}$</div>


title: in-a-triangle-mathrmabc-angle-mathrmc90circ-then-fraca2b2a2b2-is-equal-to
slug: in-a-triangle-mathrmabc-angle-mathrmc90circ-then-fraca2b2a2b2-is-equal-to-54392
tags: ['Mathematics', 'Heights and Distances', 'BITSAT', 'BITSAT 2009']
source_db: MarksBatch2_P1.db
question:
<div class="question">In a triangle \(\mathrm{ABC}, \angle \mathrm{C}=90^{\circ}\), then \(\frac{a^2-b^2}{a^2+b^2}\) is equal to :</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">\(\sin (\mathrm{A}+\mathrm{B})\)</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">\(\sin (A-B)\)</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">\(\cos (\mathrm{A}+\mathrm{B})\)</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">\(\sin \left(\frac{\mathrm{A}-\mathrm{B}}{2}\right)\)</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">\(\sin (A-B)\)</span> </div>

answer:
<div class="solution">\(\begin{aligned}<br/>&amp; \mathrm{A}+\mathrm{B}=180^{\circ}-\mathrm{C}=90^{\circ} \\<br/>&amp; \mathrm{a}=2 \mathrm{R} \sin \mathrm{A}, b=2 \mathrm{R} \sin \mathrm{B}, c=2 \mathrm{R} \sin \mathrm{C} \\<br/>&amp; \begin{array}{l}<br/>\therefore \frac{a^2-b^2}{a^2+b^2}=\frac{\sin ^2 \mathrm{~A}-\sin ^2 \mathrm{~B}}{\sin ^2 \mathrm{~A}+\sin ^2 \mathrm{~B}} \\<br/>\quad=\frac{\sin (\mathrm{A}+\mathrm{B}) \sin (\mathrm{A}-\mathrm{B})}{\sin ^2 \mathrm{~A}+\sin ^2\left(90^{\circ}-\mathrm{A}\right)} \\<br/>{\left[\because \mathrm{A}+\mathrm{B}=90^{\circ}\right]} \\<br/>=\frac{\sin 90^{\circ} \sin (\mathrm{A}-\mathrm{B})}{\sin ^2 \mathrm{~A}+\cos ^2 \mathrm{~A}}=\sin (\mathrm{A}-\mathrm{B})<br/>\end{array}<br/>\end{aligned}\)</div>


tags: ['Physics', 'Wave Optics', 'BITSAT', 'BITSAT 2015']
source_db: MarksBatch2_P1.db
title:
in-a-ydse-the-light-of-wavelength-5000-a-is-used-which-emerges-in-phase-from-two-slits-a-distance-d-3-1-0-7-m-apart-atransparent-sheet-of-thickness-t-

slug:
in-a-ydse-the-light-of-wavelength-5000-a-is-used-which-emerges-in-phase-from-two-slits-a-distance-d-3-1-0-7-m-apart-atransparent-sheet-of-thickness-t-92952

question:
<div class="question">In a YDSE, the light of wavelength $\lambda=5000 Ã$ is used, which emerges in phase from two slits a distance $\mathrm{d}=3 \times 10^{-7} \mathrm{~m}$ apart. Atransparent sheet of thickness $\mathrm{t}=1.5 \times 10^{-7} \mathrm{~m}$ refractive index $\mu=1.17$ is placed over one of the slits. what is the new angular position of the central maxima of the interference pattern, from the centre of the screen? Find the value of $y$.
<br/><img src=" data:image/png;base64,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"/></div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$4.9^{\circ}$ and $\frac{D(\mu-1) t}{2 d}$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$4.9^{\circ}$ and $\frac{\mathrm{D}(\mu-1) \mathrm{t}}{\mathrm{d}}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$3.9^{\circ}$ and $\frac{D(\mu+1) t}{d}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$2.9^{\circ}$ and $\frac{2 \mathrm{D}(\mu+1) \mathrm{t}}{\mathrm{d}}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$4.9^{\circ}$ and $\frac{\mathrm{D}(\mu-1) \mathrm{t}}{\mathrm{d}}$</span> </div>

answer:
<div class="solution">The path difference when transparent sheet is introduced $\Delta x=(\mu-1) t$
<br/> <br/>If the central maxima occupies position of nth fringe, then $(\mu-1) \mathrm{t}=\mathrm{n} \lambda=\mathrm{d} \sin \theta$
<br/> <br/>$$
<br/> <br/>\Rightarrow \sin \theta=\frac{(\mu-1) \mathrm{t}}{\mathrm{d}}=\frac{(1.17-1) \times 1.5 \times 10^{-7}}{3 \times 10^{-7}}=0.085
<br/> <br/>$$
<br/> <br/>Therefore, angular position of central maxima
<br/> <br/>$$
<br/> <br/>\theta=\sin ^{-1}(0.085)=4.88^{\circ} \approx 4.9
<br/> <br/>$$
<br/> <br/>For small angles, $\sin \theta=\theta=\tan \theta$
<br/> <br/>$$
<br/> <br/>\Rightarrow \quad \tan \theta=\frac{y}{D}
<br/> <br/>$$
<br/> <br/>$\therefore \quad \frac{\mathrm{y}}{\mathrm{D}}=\frac{(\mu-1) \mathrm{t}}{\mathrm{d}} \Rightarrow \mathrm{y}=\frac{\mathrm{D}(\mu-1) \mathrm{t}}{\mathrm{d}}$</div>


tags: ['Physics', 'Wave Optics', 'BITSAT', 'BITSAT 2022']
source_db: MarksBatch2_P1.db
title:
in-a-youngs-double-slit-arrangement-frings-are-produced-using-light-of-wavelength-4000-a-one-slit-is-covered-by-a-thin-plate-of-glass-of-refractive-in

slug:
in-a-youngs-double-slit-arrangement-frings-are-produced-using-light-of-wavelength-4000-a-one-slit-is-covered-by-a-thin-plate-of-glass-of-refractive-in-84378

question:
<div class="question">In a young's double slit arrangement frings are produced using light of wavelength $4000 Ã$. One slit is covered by a thin plate of glass of refractive index $1.4$ and the other with another glass plate of same thickness but of refractive index 1.7. By doing so the central bright shifts to original sixth fringe from centre. Thickness of glass plate is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"> <br/>$2 \mu \mathrm{m}$
<br/></span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data"> <br/>$8 \mu \mathrm{m}$
<br/></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data"> <br/>$11 \mu \mathrm{m}$
<br/></span> </li><li class=""> <span class="option-label">D</span> <span class="option-data"> <br/>$16 \mu \mathrm{m}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"> <br/>$8 \mu \mathrm{m}$
<br/></span> </div>

answer:
<div class="solution">As, shift $\Delta x=\frac{\beta}{\lambda}(\mu-1) t$
<br/><img src="https://cdn-question-pool.getmarks.app/pyq/bitsat/oDxtwpHhNzALTMXMkFLQwGcfVoCSMecfGp-bCS6zZZM.original.fullsize.png"/><br/> <br/>Shift due to one plate $\Delta x_1=\frac{\beta}{\lambda}\left(\mu_1-1\right) t$
<br/>Shift due to another path $\Delta x_2=\frac{\beta}{\lambda}\left(\mu_2-1\right) t$
<br/>Net shift $\Delta x=\Delta x_2-\Delta x_1=\frac{\beta}{\lambda}\left(\mu_2-\mu_1\right) t$
<br/>Also, it is given that $\Delta x=6 \beta$
<br/>$$
<br/>\begin{aligned}
<br/>&amp; \text { Hence, } 6 \beta=\frac{\beta}{\lambda}\left(\mu_1-\mu_2\right) t \\
<br/>&amp; \Rightarrow \quad t=\frac{6 \lambda}{\left(\mu_1-\mu_2\right)}=\frac{6 \times 4000}{(1.7-1.4)}-80000 Ã \\
<br/>&amp; \Rightarrow \quad t=8 \times 10^{-6}=8 \mu \mathrm{m}
<br/>\end{aligned}
<br/>$$</div>


tags: ['Physics', 'Wave Optics', 'BITSAT', 'BITSAT 2012']
source_db: MarksBatch2_P1.db
title:
in-a-youngs-double-slit-experiment-the-angular-width-of-a-fringe-formed-on-a-distant-screen-is-1-the-wavelength-fo-the-light-used-is-6280-a-what-is-th

slug:
in-a-youngs-double-slit-experiment-the-angular-width-of-a-fringe-formed-on-a-distant-screen-is-1-the-wavelength-fo-the-light-used-is-6280-a-what-is-th-40464

question:
<div class="question">In a Young's double slit experiment the angular width of a fringe formed on a distant screen is $1^{\circ}$. The wavelength fo the light used is $6280 \AA$. What is the distance between the two coherent sources?</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$0.036 \mathrm{~mm}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$0.12 \mathrm{~mm}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$6 \mathrm{~mm}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$4 \mathrm{~mm}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$0.036 \mathrm{~mm}$</span> </div>

answer:
<div class="solution">The angular fringe width is given by $\alpha=\frac{\lambda}{\mathrm{d}}$
<br/>where $\lambda$ is wavelength and $\mathrm{d}$ is the distance between two coherent sources. Thus
<br/>$\begin{array}{l}
<br/>\mathrm{d}=\frac{\lambda}{\alpha} \\
<br/>\text { Given, } \lambda=6280 \AA, \alpha=1^{\circ}=\frac{\pi}{180} \mathrm{radian}
<br/>\end{array}$
<br/>Thus $\mathrm{d}=\frac{6280 \times 10^{-10}}{3.14} \times 180$
<br/>$=3.6 \times 10^{-5} \mathrm{~m}=0.036 \mathrm{~mm}$</div>


tags: ['Physics', 'Wave Optics', 'BITSAT', 'BITSAT 2023 (Memory Based Paper 1)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">3</span> </div>
source_db: MarksBatch2_P1.db
title:
in-a-youngs-double-slit-experiment-the-intensities-at-two-points-for-the-path-difference-4-and-3-being-the-wavelength-of-light-used-are-i-1-and-i-2-re-1

slug:
in-a-youngs-double-slit-experiment-the-intensities-at-two-points-for-the-path-difference-4-and-3-being-the-wavelength-of-light-used-are-i-1-and-i-2-re-1-74692

question:
<div class="question">In a Young's double slit experiment, the intensities at two points, for the path difference $\frac{\lambda}{4}$ and $\frac{\lambda}{3}$ ( $\lambda$ being the wavelength of light used) are $I_1$ and $I_2$ respectively. If $I_0$ denotes the intensity produced by each one of the individual slits, then $\frac{\mathrm{I}_1+\mathrm{I}_2}{\mathrm{I}_0}=\ldots \ldots$</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">3</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">5</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">7</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">10</span> </li> </ul>

answer:
<div class="solution">Resultant intensity in Young's double slit experiment<br/>$\mathrm{I}=4 \mathrm{I}_0 \cos ^2\left(\frac{\Delta \phi}{2}\right)$<br/>For path difference $\frac{\lambda}{4}$ phase difference,<br/>$\Delta \phi=\frac{2 \pi}{\lambda} \times \frac{\lambda}{4}=\frac{\pi}{4}$<br/>$\therefore \mathrm{I}_1=4 \mathrm{I}_0 \cos ^2\left(\frac{\pi}{4}\right)=2 \mathrm{I}_0$<br/>For path difference $\frac{\lambda}{3}$<br/>$I_2=4 I_0 \cos ^2\left(\frac{2 \pi}{\lambda} \times \frac{\lambda}{3}\right)=I_0$<br/>$\therefore \frac{\mathrm{I}_1+\mathrm{I}_2}{\mathrm{I}_0}=3$</div>


tags: ['Chemistry', 'Thermodynamics (C)', 'BITSAT', 'BITSAT 2022']
source_db: MarksBatch2_P1.db
title:
in-an-adiabatic-process-no-transfer-of-heat-takes-place-between-system-and-surrounding-choose-the-correct-option-for-free-expansion-of-an-ideal-gas-un

slug:
in-an-adiabatic-process-no-transfer-of-heat-takes-place-between-system-and-surrounding-choose-the-correct-option-for-free-expansion-of-an-ideal-gas-un-36712

question:
<div class="question">In an adiabatic process, no transfer of heat takes place between system and surrounding. Choose the correct option for free expansion of an ideal gas under adiabatic condition from the following.</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"> <br/>$q=0, \Delta T \neq 0, W=0$
<br/></span> </li><li class=""> <span class="option-label">B</span> <span class="option-data"> <br/>$q \neq 0, \Delta T=0, W=0$
<br/></span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data"> <br/>$q=0, \Delta T=0, W=0$
<br/></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data"> <br/>$q=0, \Delta T &lt; 0, W \neq 0$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"> <br/>$q=0, \Delta T=0, W=0$
<br/></span> </div>

answer:
<div class="solution">Free expansion, $W=0$
<br/>Adiabatic process, $q=0$
<br/>$\Delta U=q+W=0$, this means that internal energy remains constant. Therefore, $\Delta T=0$ in ideal gas there is no intermolecular attraction.
<br/>Hence, when such a gas expands under adiabatic conditions into a vacuum, no heat is absorbed or evolved, since no external work is done to separate the molecules.</div>


title: in-an-adiabatic-process-the-pressure-is-increased-by-3-2-if-2-3-then-the-volume-decreases-by-nearly
tags: ['Physics', 'Thermodynamics', 'BITSAT', 'BITSAT 2020']
source_db: MarksBatch2_P1.db
slug:
in-an-adiabatic-process-the-pressure-is-increased-by-3-2-if-2-3-then-the-volume-decreases-by-nearly-41325

question:
<div class="question">In an adiabatic process, the pressure is increased by $\frac{2}{3} \% .$ If $\gamma=\frac{3}{2},$ then the volume decreases by nearly</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$\frac{4}{9} \%$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\frac{2}{3} \%$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$1 \%$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\frac{9}{4} \%$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\frac{4}{9} \%$</span> </div>

answer:
<div class="solution">$\mathrm{PV}^{3 / 2}=\mathrm{K}$
<br/> <br/>$\log \mathrm{P}+\frac{3}{2} \log \mathrm{V}=\log \mathrm{K}$
<br/> <br/>$\frac{\Delta \mathrm{P}}{\mathrm{P}}+\frac{3}{2} \frac{\Delta \mathrm{V}}{\mathrm{V}}=0$
<br/> <br/>$\frac{\Delta \mathrm{V}}{\mathrm{V}}=-\frac{2}{3} \frac{\Delta \mathrm{P}}{\mathrm{P}}$ or $\frac{\Delta \mathrm{V}}{\mathrm{V}}=\left(-\frac{2}{3}\right)\left(\frac{2}{3}\right)=-\frac{4}{9}$</div>


tags: ['Chemistry', 'Structure of Atom', 'BITSAT', 'BITSAT 2018']
source_db: MarksBatch2_P1.db
title:
in-an-atom-an-electron-is-moving-with-a-speed-of-600-m-s-with-an-accuracy-of-0-005-certainty-with-which-the-position-of-the-electron-can-be-located-is

slug:
in-an-atom-an-electron-is-moving-with-a-speed-of-600-m-s-with-an-accuracy-of-0-005-certainty-with-which-the-position-of-the-electron-can-be-located-is-60177

question:
<div class="question">In an atom, an electron is moving with a speed of <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>600</mn><mo> </mo><mi mathvariant="normal">m</mi><mo>/</mo><mi mathvariant="normal">s</mi></math> with an accuracy of <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo>.</mo><mn>005</mn><mo>%</mo></math>. Certainty with which the position of the electron can be located is (<math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">h</mi><mo>=</mo><mn>6</mn><mo>.</mo><mn>6</mn><mo>×</mo><msup><mn>10</mn><mrow><mo>-</mo><mn>34</mn></mrow></msup><mo> </mo><mi>kg</mi><mo> </mo><msup><mi mathvariant="normal">m</mi><mn>2</mn></msup><mo> </mo><msup><mi mathvariant="normal">s</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup></math>, mass of electron,  <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi mathvariant="normal">m</mi><mi mathvariant="normal">e</mi></msub><mo>=</mo><mn>9</mn><mo>.</mo><mn>1</mn><mo>×</mo><msup><mn>10</mn><mrow><mo>-</mo><mn>31</mn></mrow></msup><mo> </mo><mi>kg</mi></math>)</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn><mo>.</mo><mn>52</mn><mo>×</mo><msup><mn>10</mn><mrow><mo>-</mo><mn>4</mn></mrow></msup><mo> </mo><mi mathvariant="normal">m</mi></math></span> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>5</mn><mo>.</mo><mn>10</mn><mo>×</mo><msup><mn>10</mn><mrow><mo>-</mo><mn>3</mn></mrow></msup><mo> </mo><mi mathvariant="normal">m</mi></math></span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn><mo>.</mo><mn>92</mn><mo>×</mo><msup><mn>10</mn><mrow><mo>-</mo><mn>3</mn></mrow></msup><mo> </mo><mi mathvariant="normal">m</mi></math></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>3</mn><mo>.</mo><mn>84</mn><mo>×</mo><msup><mn>10</mn><mrow><mo>-</mo><mn>3</mn></mrow></msup><mo> </mo><mi mathvariant="normal">m</mi></math></span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn><mo>.</mo><mn>92</mn><mo>×</mo><msup><mn>10</mn><mrow><mo>-</mo><mn>3</mn></mrow></msup><mo> </mo><mi mathvariant="normal">m</mi></math></span> </div>

answer:
<div class="solution"><p>Given that velocity of electron, <math><mi>v</mi><mo>=</mo><mn>600</mn><mi> </mi><mrow><mrow><mi>m</mi></mrow><mo>/</mo><mrow><mi>s</mi></mrow></mrow></math></p>Accuracy of velocity <math><mo>=</mo><mn>0.005</mn><mi>%</mi></math><br/><br/>According to Heisenberg's uncertainty principle<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mi>Δx</mi></mfenced><mo>.</mo><mfenced><mi>mΔv</mi></mfenced><mo>=</mo><mfrac><mi mathvariant="normal">h</mi><mrow><mn>4</mn><mi mathvariant="normal">π</mi></mrow></mfrac></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>∆</mo><mi mathvariant="normal">v</mi><mo>=</mo><mn>0</mn><mo>.</mo><mn>005</mn><mo>%</mo></math> or <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>600</mn><mo> </mo><mi mathvariant="normal">m</mi><mo> </mo><msup><mi mathvariant="normal">s</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfrac><mrow><mn>600</mn><mo>×</mo><mn>0</mn><mo>.</mo><mn>005</mn></mrow><mn>100</mn></mfrac><mo>=</mo><mn>0</mn><mo>.</mo><mn>03</mn></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>∆</mo><mi mathvariant="normal">x</mi><mo>×</mo><mn>9</mn><mo>.</mo><mn>1</mn><mo>×</mo><msup><mn>10</mn><mrow><mo>-</mo><mn>31</mn></mrow></msup><mo>×</mo><mn>0</mn><mo>.</mo><mn>03</mn><mo>=</mo><mfrac><mrow><mn>6</mn><mo>.</mo><mn>6</mn><mo>×</mo><msup><mn>10</mn><mrow><mo>-</mo><mn>34</mn></mrow></msup></mrow><mrow><mn>4</mn><mo>×</mo><mn>3</mn><mo>.</mo><mn>14</mn></mrow></mfrac></math></p><p>Hence, <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>∆</mo><mi>x</mi><mo>=</mo><mfrac><mrow><mn>6</mn><mo>.</mo><mn>6</mn><mo>×</mo><msup><mn>10</mn><mrow><mo>-</mo><mn>34</mn></mrow></msup></mrow><mrow><mn>4</mn><mo>×</mo><mn>3</mn><mo>.</mo><mn>14</mn><mo>×</mo><mn>0</mn><mo>.</mo><mn>03</mn><mo>×</mo><mn>9</mn><mo>.</mo><mn>1</mn><mo>×</mo><msup><mn>10</mn><mrow><mo>-</mo><mn>31</mn></mrow></msup></mrow></mfrac></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>1</mn><mo>.</mo><mn>92</mn><mo>×</mo><msup><mn>10</mn><mrow><mo>-</mo><mn>3</mn></mrow></msup><mo> </mo><mi mathvariant="normal">m</mi></math></p></div>


title: in-an-equilateral-triangle-the-in-radius-circumradius-and-one-of-the-exradii-are-in-the-ratio
slug: in-an-equilateral-triangle-the-in-radius-circumradius-and-one-of-the-exradii-are-in-the-ratio-69177
tags: ['Mathematics', 'Properties of Triangles', 'BITSAT', 'BITSAT 2011']
source_db: MarksBatch2_P1.db
question:
<div class="question">In an equilateral triangle, the in radius, circumradius and one of the ex-radii are in the ratio</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">\(2: 3: 5\)</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">\(1: 2: 3\)</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">\(3: 7: 9\)</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">\(3: 7: 9\)</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">\(1: 2: 3\)</span> </div>

answer:
<div class="solution">We have \(\Delta=\frac{\sqrt{3}}{4} \mathrm{a}^2, \mathrm{~s}=\frac{3 \mathrm{a}}{2}\)<br/>\(\therefore \quad r=\frac{\Delta}{\mathrm{s}}=\frac{\mathrm{a}}{2 \sqrt{3}}, \mathrm{R}=\frac{\mathrm{abc}}{4 \Delta}=\frac{\mathrm{a}^3}{\sqrt{3} \mathrm{a}^2}=\frac{\mathrm{a}}{\sqrt{3}}\) and \(r_1=\frac{\Delta}{s-a}=\frac{\sqrt{3} / 4 a^2}{a / 2}=\frac{\sqrt{3}}{2} a\)<br/>Hence, \(\mathrm{r}: \mathrm{R}: \mathrm{r}_1=\frac{\mathrm{a}}{2 \sqrt{3}}: \frac{\mathrm{a}}{\sqrt{3}}: \frac{\sqrt{3}}{2} \mathrm{a}=1: 2: 3\)</div>


tags: ['Physics', 'Dual Nature of Matter', 'BITSAT', 'BITSAT 2012']
source_db: MarksBatch2_P1.db
title:
in-an-experiment-on-photoelectric-effect-photons-of-wavelength-300-nm-eject-electrons-from-a-metal-of-work-function-225-ev-a-photon-of-energy-equal-to

slug:
in-an-experiment-on-photoelectric-effect-photons-of-wavelength-300-nm-eject-electrons-from-a-metal-of-work-function-225-ev-a-photon-of-energy-equal-to-43628

question:
<div class="question">In an experiment on photoelectric effect photons of wavelength $300 \mathrm{nm}$ eject electrons from a metal of work function $2.25 \mathrm{eV}$. A photon of energy equal to that of the most energetic electron corresponds to the following transition in the hydrogen atom:</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$n=2$ to $n=1$ state</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$n=3$ to $n=1$ state</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$n=3$ to $n=2$ state</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$n=4$ to $n=3$ state</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$n=3$ to $n=2$ state</span> </div>

answer:
<div class="solution">Given $\phi=2.25 \mathrm{eV}$
<br/>$$
<br/>\lambda=300 \times 100^{-3} \mathrm{~m}
<br/>$$
<br/>Energy of the given photon,
<br/>$$
<br/>\mathrm{E}=\mathrm{hv}
<br/>$$
<br/>$$
<br/>=\frac{\mathrm{hc}}{\lambda}=\frac{663 \times 10^{-34} \times 3 \times 10^8}{300 \times 10^{-9}}=6.63 \times 10^{-19} \mathrm{~J}=4.13 \mathrm{eV}
<br/>$$
<br/>Now,
<br/>$$
<br/>\mathrm{E}=\frac{\mathrm{hc}}{\lambda}-\phi=4.137-2.25=1.88 \mathrm{eV} \rightarrow(1)
<br/>$$
<br/>So, $1.88 \mathrm{eV}$ energy is used to jump from one orbit to another orbit by electron. Therefore, energy of different orbital of hydrogen
<br/>$$
<br/>\begin{array}{rrrr}
<br/>\Rightarrow \mathrm{n}=1 &amp; 2 &amp; 3 &amp; 4 \\
<br/>\mathrm{E}=-13.6 \mathrm{ev} &amp; -3.4 &amp; -1.51 &amp; -0.85
<br/>\end{array}
<br/>$$
<br/>And from statement (1),
<br/>$$
<br/>1.18=E_i-E_f
<br/>$$
<br/>$\left\{E_i \&amp; E_f \quad\right.$ are energy of initial and final orbit $\}$ And also from the above table, we can observe that,
<br/>$$
<br/>E_{\mathrm{i}}-\mathrm{E}_{\mathrm{f}}=(-1.51)-(-3.4)=1.89 \mathrm{eV}
<br/>$$
<br/>$$
<br/>=\mathrm{E}_2-\mathrm{E}_3
<br/>$$
<br/>That the electron jumps from 3rd to 2nd orbit. Therefore, with one photon of 3oonm wavelength hydrogen electron can jump from 3 to 2 .</div>


tags: ['Physics', 'Electromagnetic Induction', 'BITSAT', 'BITSAT 2023 (Memory Based Paper 1)']
source_db: MarksBatch2_P1.db
title:
in-an-lc-oscillator-if-values-of-inductance-and-capacitance-become-twice-and-eight-times-respectively-then-the-resonant-frequency-of-oscillator-become-1

slug:
in-an-lc-oscillator-if-values-of-inductance-and-capacitance-become-twice-and-eight-times-respectively-then-the-resonant-frequency-of-oscillator-become-1-90815

question:
<div class="question">In an LC oscillator, if values of inductance and capacitance become twice and eight times, respectively, then the resonant frequency of oscillator becomes $\mathrm{x}$ times its initial resonant frequency $\omega_0$. The value of $x$ is:</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$1 / 4$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">16</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$1 / 16$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">4</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$1 / 4$</span> </div>

answer:
<div class="solution">The resonance frequency of LC oscillations circuit is<br/>$\omega=\frac{1}{\sqrt{\mathrm{L}^{\prime} \mathrm{C}^{\prime}}} \Rightarrow \mathrm{L}^{\prime} \rightarrow 2 \mathrm{~L}$<br/>$\mathrm{C}^{\prime} \rightarrow 8 \mathrm{C}$<br/>$\omega=\frac{1}{\sqrt{2 \mathrm{~L} \times 8 \mathrm{C}}}=\frac{1}{4 \sqrt{\mathrm{LC}}} 9 \omega_0=\frac{1}{\sqrt{\mathrm{LC}}}$<br/>$\omega=\frac{\omega_0}{4}$ So, $x=\frac{1}{4}$</div>


tags: ['Physics', 'Nuclear Physics', 'BITSAT', 'BITSAT 2015']
source_db: MarksBatch2_P1.db
title:
in-an-ore-containing-uranium-the-ratio-of-u-238-to-pb-206-is-3-calculate-the-ge-of-the-ore-assuming-that-all-the-lead-present-in-the-ore-is-the-final-

slug:
in-an-ore-containing-uranium-the-ratio-of-u-238-to-pb-206-is-3-calculate-the-ge-of-the-ore-assuming-that-all-the-lead-present-in-the-ore-is-the-final-48910

question:
<div class="question">In an ore containing uranium, the ratio of $U^{238}$ to $\mathrm{Pb}^{206}$ is 3. Calculate the Ã ge of the ore, assuming that all the lead present in the ore is the final stable product of $U^{238}$. Take the half-life of $U^{238}$ to be $4.5 \times 10^{9} \mathrm{yr}$.</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$1.6 \times 19^{3} \mathrm{yr}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$1.5 \times 10^{4} \mathrm{yr}$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$1.867 \times 10^{9} \mathrm{yr}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$2 \times 10^{5} \mathrm{yr}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$1.867 \times 10^{9} \mathrm{yr}$</span> </div>

answer:
<div class="solution">Let the initial mass of uranium be $\mathrm{M}_{0}$ Final mass of uranium after time $\mathrm{t}$
<br/> <br/>$$
<br/> <br/>\mathrm{M}=\frac{3}{4} \mathrm{M}_{0}
<br/> <br/>$$
<br/> <br/>According to the law of radioactive disintegration.
<br/> <br/>$$
<br/> <br/>\begin{array}{l}
<br/> <br/>\frac{\mathrm{M}}{\mathrm{M}_{0}}=\left(\frac{1}{2}\right)^{\mathrm{t} / \mathrm{T}} \Rightarrow \frac{\mathrm{M}_{0}}{\mathrm{M}}=(2)^{\mathrm{t} / \mathrm{T}} \\
<br/> <br/>\therefore \log _{10}\left(\frac{\mathrm{M}_{0}}{\mathrm{M}}\right)=\frac{\mathrm{t}}{\mathrm{T}} \log _{10}(2) \\
<br/> <br/>\mathrm{t}=\mathrm{T} \frac{\log _{10}\left(\frac{\mathrm{M}_{0}}{\mathrm{M}}\right)}{\log _{10}(2)}=\frac{\mathrm{T} \log _{10}\left(\frac{4}{3}\right)}{\log _{10}(2)} \\
<br/> <br/>=\frac{\mathrm{T} \log _{10}(1.333)}{\log _{10}(2)}=4.5 \times 10^{9}\left(\frac{0.1249}{0.3010}\right) \\
<br/> <br/>\Rightarrow \mathrm{t}=1.867 \times 10^{9} \mathrm{yr} .
<br/> <br/>\end{array}
<br/> <br/>$$</div>


tags: ['Physics', 'Electrostatics', 'JEE Main']
source_db: MarksBatch2_P1.db
title:
in-an-uniformly-charged-sphere-of-total-charge-q-and-radius-r-the-electric-field-e-is-plotted-as-function-of-distance-from-the-centre-the-graph-which--1

slug:
in-an-uniformly-charged-sphere-of-total-charge-q-and-radius-r-the-electric-field-e-is-plotted-as-function-of-distance-from-the-centre-the-graph-which-1-16426

question:
<div class="question">In an uniformly charged sphere of total charge Q and radius $\mathrm{R},$ the electric field $\mathrm{E}$ is plotted as function of distance from the centre. The graph which would correspond to the above will be</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><img src="https://cdn-question-pool.getmarks.app/pyq/bitsat/5g3wlwkjOFFNVL-73z9d5nXenCg3gegSoxB_eiGjKwk.original.fullsize.png"/></span> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><img src="https://cdn-question-pool.getmarks.app/pyq/bitsat/PGS4imYljxacnBqSujsGbmu7gGwhjwJ_3WMkmF9ZTds.original.fullsize.png"/></span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data"><img src="https://cdn-question-pool.getmarks.app/pyq/bitsat/RRtdjCK5zmkxA9ivUBZaOkoE6SWm_uIQWR4WEK7TkEs.original.fullsize.png"/></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><img src="https://cdn-question-pool.getmarks.app/pyq/bitsat/L1H63FoWiLT_MlmJsC5BDH4ro2ynZxgmRpmgvIq6i4k.original.fullsize.png"/></span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><img src="https://cdn-question-pool.getmarks.app/pyq/bitsat/RRtdjCK5zmkxA9ivUBZaOkoE6SWm_uIQWR4WEK7TkEs.original.fullsize.png"/></span> </div>

answer:
<div class="solution">Electric field inside the uniformly charged sphere varies linearly, $E=\frac{k Q}{R^{3}} r,(r \leq R)$ while outside the sphere, it varies as inverse<br/>square of distance, $E=\frac{k Q}{r^{2}} ;(r \geq R)$ which is correctly represented in option (c).</div>


tags: ['Physics', 'Electrostatics', 'BITSAT', 'BITSAT 2011']
source_db: MarksBatch2_P1.db
title:
in-an-uniformly-charged-sphere-of-total-charge-q-and-radius-r-the-electric-field-e-is-plotted-as-function-of-distance-from-the-centre-the-graph-which-

slug:
in-an-uniformly-charged-sphere-of-total-charge-q-and-radius-r-the-electric-field-e-is-plotted-as-function-of-distance-from-the-centre-the-graph-which-19822

question:
<div class="question">In an uniformly charged sphere of total charge Q and radius $\mathrm{R},$ the electric field $\mathrm{E}$ is plotted as function of distance from the centre. The graph which would correspond to the above will be</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><img src="https://cdn-question-pool.getmarks.app/pyq/bitsat/5g3wlwkjOFFNVL-73z9d5nXenCg3gegSoxB_eiGjKwk.original.fullsize.png"/></span> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><img src="https://cdn-question-pool.getmarks.app/pyq/bitsat/PGS4imYljxacnBqSujsGbmu7gGwhjwJ_3WMkmF9ZTds.original.fullsize.png"/></span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data"><img src="https://cdn-question-pool.getmarks.app/pyq/bitsat/RRtdjCK5zmkxA9ivUBZaOkoE6SWm_uIQWR4WEK7TkEs.original.fullsize.png"/></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><img src="https://cdn-question-pool.getmarks.app/pyq/bitsat/L1H63FoWiLT_MlmJsC5BDH4ro2ynZxgmRpmgvIq6i4k.original.fullsize.png"/></span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><img src="https://cdn-question-pool.getmarks.app/pyq/bitsat/RRtdjCK5zmkxA9ivUBZaOkoE6SWm_uIQWR4WEK7TkEs.original.fullsize.png"/></span> </div>

answer:
<div class="solution">Electric field inside the uniformly charged sphere varies linearly, $E=\frac{k Q}{R^{3}} r,(r \leq R)$ while outside the sphere, it varies as inverse<br/>square of distance, $E=\frac{k Q}{r^{2}} ;(r \geq R)$ which is correctly represented in option (c).</div>


title: in-case-of-a-forced-vibration-the-resonance-wave-becomes-very-sharp-when-the
slug: in-case-of-a-forced-vibration-the-resonance-wave-becomes-very-sharp-when-the-71785
tags: ['Physics', 'Oscillations', 'BITSAT', 'BITSAT 2009']
answer: <div class="solution">No Solution available</div>
source_db: MarksBatch2_P1.db
question:
<div class="question">In case of a forced vibration, the resonance wave becomes very sharp when the</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">restoring force is small</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">applied periodic force is small</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">quality factor is small</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">damping force is small</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">damping force is small</span> </div>


tags: ['Physics', 'Semiconductors', 'BITSAT', 'BITSAT 2015']
source_db: MarksBatch2_P1.db
title:
in-ce-transistor-amplifier-the-audio-signal-voltage-across-the-collector-resistance-of-2-k-is-2-v-if-the-base-resistance-is-1-k-and-the-current-amplif

slug:
in-ce-transistor-amplifier-the-audio-signal-voltage-across-the-collector-resistance-of-2-k-is-2-v-if-the-base-resistance-is-1-k-and-the-current-amplif-45733

question:
<div class="question">In CE transistor amplifier, the audio signal voltage across the collector resistance of $2 \mathrm{k} \Omega$ is $2 \mathrm{~V}$. If the base resistance is $1 \mathrm{k} \Omega$ and the current amplification of the transistor is $100,$ the input signal voltage is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$2 \mathrm{mV}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$3 \mathrm{mV}$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$10 \mathrm{mV}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$0.1 \mathrm{mV}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$10 \mathrm{mV}$</span> </div>

answer:
<div class="solution">Given : Voltage across the collector $\mathrm{V}_{0}=2$ $\mathrm{V} ;$ collector resistance, $\mathrm{R}_{\mathrm{c}}=2 \times 10^{3} \Omega ;$ Base resistance $\mathrm{R}_{\mathrm{B}}=1 \times 10^{3} \Omega ;$ Input signal voltage, $\mathrm{V}_{\mathrm{i}}=?$
<br/> <br/><img src="https://cdn-question-pool.getmarks.app/pyq/bitsat/l0NBBmnfAsxTTW4WWPBulMoH88i0MrGxL2_ZPZENPtU.original.fullsize.png"/><br/> <br/> <br/>$$
<br/> <br/>\begin{array}{l}
<br/> <br/>\mathrm{V}_{0}=\mathrm{I}_{\mathrm{C}} \mathrm{R}_{\mathrm{C}}=2 \\
<br/> <br/>\Rightarrow \quad \mathrm{I}_{\mathrm{C}}=\frac{2}{2 \times 10^{3}}=10^{-3} \mathrm{~A} \\
<br/> <br/>\text { Current gain } \alpha=\frac{\mathrm{I}_{\mathrm{C}}}{\mathrm{I}_{\mathrm{B}}}=100 \\
<br/> <br/>\Rightarrow \quad \mathrm{I}_{\mathrm{B}}=\frac{\mathrm{I}_{\mathrm{C}}}{100}=\frac{10^{-3}}{100}=10^{-5} \mathrm{~A} \\
<br/> <br/>\quad \mathrm{~V}_{\mathrm{i}}=\mathrm{R}_{\mathrm{B}} \mathrm{I}_{\mathrm{B}} \Rightarrow \mathrm{V}_{\mathrm{i}}=1 \times 10^{3} \times 10^{-5} \\
<br/> <br/>\Rightarrow \quad \mathrm{V}_{\mathrm{i}}=10^{-2} \mathrm{~V} \Rightarrow \mathrm{V}_{\mathrm{i}}=10 \mathrm{mV}
<br/> <br/>\end{array}
<br/> <br/>$$</div>


tags: ['Physics', 'Laws of Motion', 'JEE Main']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">0.4</span> </div>
source_db: MarksBatch2_P1.db
title:
in-figure-two-blocks-are-separated-by-a-uniform-strut-attached-to-each-block-with-frictionless-pins-block-a-weighs-400-n-block-b-weighs-300-n-and-the--1

slug:
in-figure-two-blocks-are-separated-by-a-uniform-strut-attached-to-each-block-with-frictionless-pins-block-a-weighs-400-n-block-b-weighs-300-n-and-the-1-83196

question:
<div class="question">In figure, two blocks are separated by a uniform strut attached to each block with frictionless pins. Block $A$ weighs $400 \mathrm{~N}$, block $B$ weighs $300 \mathrm{~N}$, and the strut $A B$ weigh $200 \mathrm{~N}$. If $\mu=0.25$ under $B,$ determine the minimum coefficient of friction under $A$ to prevent motion.
<br/> <br/><img src="https://cdn-question-pool.getmarks.app/pyq/bitsat/omLMLJPKIGTPBb7aPMA14fOOR__XOuQxMeKlfo5GtpA.original.fullsize.png"/></div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">0.4</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">0.2</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">0.8</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">0.1</span> </li> </ul>

answer:
<div class="solution">Consider FBD of structure.
<br/> <br/><img src="https://cdn-question-pool.getmarks.app/pyq/bitsat/D70XSDP5MXeZWG0aDB51W2yRmBo-308i5TSx6ME1raE.original.fullsize.png"/><br/> <br/> <br/>Applying equilibrium equations, $A v+B v=200 \mathrm{~N}$
<br/> <br/>$A_{H}=B_{H} \ldots$ (ii)
<br/> <br/>From FBD of block $B$,
<br/> <br/><img src="https://cdn-question-pool.getmarks.app/pyq/bitsat/QIqyq69X0iooZNZjCpfl1kKvp3SRpHXW_q1sc-urbU0.original.fullsize.png"/><br/> <br/> <br/>$B_{H}+F_{B} \cos 60^{\circ}-N_{B} \sin 60^{\circ}=0$
<br/> <br/>$N_{B} \cos 60^{\circ}-B_{V}-300+F_{B} \sin 60^{\circ}=0$
<br/> <br/>$F_{B}=0.25 N_{B}$
<br/> <br/>$B_{H}-0.74 N_{B}=0$
<br/> <br/>(iii) $-B_{V}+0.71 N_{B}=300$
<br/> <br/>FBD of block $A$
<br/> <br/><img src="https://cdn-question-pool.getmarks.app/pyq/bitsat/Asbe3lJiSjlaDYPfN8mCE5fm0x-35whtvcEhyGJXRy4.original.fullsize.png"/><br/> <br/> <br/>$F_{A}-A_{H}=0$
<br/> <br/>$N_{A}-A_{V}=400 \quad \ldots$
<br/> <br/>$F_{A}=\mu_{A} N_{A}$
<br/> <br/>$\therefore \mu_{A} N_{A}-A_{H}=0 \quad \ldots$
<br/> <br/>on solving the above equation
<br/> <br/>$N_{A}=650 \mathrm{~N}, F_{A}=260 \mathrm{~N}, F_{A}=\mu_{A} N_{A}$
<br/> <br/>$\therefore \mu_{A}=\frac{260}{250}=0.4$</div>


tags: ['Physics', 'Laws of Motion', 'BITSAT', 'BITSAT 2015']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">0.4</span> </div>
source_db: MarksBatch2_P1.db
title:
in-figure-two-blocks-are-separated-by-a-uniform-strut-attached-to-each-block-with-frictionless-pins-block-a-weighs-400-n-block-b-weighs-300-n-and-the-

slug:
in-figure-two-blocks-are-separated-by-a-uniform-strut-attached-to-each-block-with-frictionless-pins-block-a-weighs-400-n-block-b-weighs-300-n-and-the-52930

question:
<div class="question">In figure, two blocks are separated by a uniform strut attached to each block with frictionless pins. Block $A$ weighs $400 \mathrm{~N}$, block $B$ weighs $300 \mathrm{~N}$, and the strut $A B$ weigh $200 \mathrm{~N}$. If $\mu=0.25$ under $B,$ determine the minimum coefficient of friction under $A$ to prevent motion.
<br/> <br/><img src="https://cdn-question-pool.getmarks.app/pyq/bitsat/omLMLJPKIGTPBb7aPMA14fOOR__XOuQxMeKlfo5GtpA.original.fullsize.png"/></div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">0.4</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">0.2</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">0.8</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">0.1</span> </li> </ul>

answer:
<div class="solution">Consider FBD of structure.
<br/> <br/><img src="https://cdn-question-pool.getmarks.app/pyq/bitsat/D70XSDP5MXeZWG0aDB51W2yRmBo-308i5TSx6ME1raE.original.fullsize.png"/><br/> <br/> <br/>Applying equilibrium equations, $A v+B v=200 \mathrm{~N}$
<br/> <br/>$A_{H}=B_{H} \ldots$ (ii)
<br/> <br/>From FBD of block $B$,
<br/> <br/><img src="https://cdn-question-pool.getmarks.app/pyq/bitsat/QIqyq69X0iooZNZjCpfl1kKvp3SRpHXW_q1sc-urbU0.original.fullsize.png"/><br/> <br/> <br/>$B_{H}+F_{B} \cos 60^{\circ}-N_{B} \sin 60^{\circ}=0$
<br/> <br/>$N_{B} \cos 60^{\circ}-B_{V}-300+F_{B} \sin 60^{\circ}=0$
<br/> <br/>$F_{B}=0.25 N_{B}$
<br/> <br/>$B_{H}-0.74 N_{B}=0$
<br/> <br/>(iii) $-B_{V}+0.71 N_{B}=300$
<br/> <br/>FBD of block $A$
<br/> <br/><img src="https://cdn-question-pool.getmarks.app/pyq/bitsat/Asbe3lJiSjlaDYPfN8mCE5fm0x-35whtvcEhyGJXRy4.original.fullsize.png"/><br/> <br/> <br/>$F_{A}-A_{H}=0$
<br/> <br/>$N_{A}-A_{V}=400 \quad \ldots$
<br/> <br/>$F_{A}=\mu_{A} N_{A}$
<br/> <br/>$\therefore \mu_{A} N_{A}-A_{H}=0 \quad \ldots$
<br/> <br/>on solving the above equation
<br/> <br/>$N_{A}=650 \mathrm{~N}, F_{A}=260 \mathrm{~N}, F_{A}=\mu_{A} N_{A}$
<br/> <br/>$\therefore \mu_{A}=\frac{260}{250}=0.4$</div>


title: in-freundlich-adsorption-isotherm-the-value-of-1-n-is
slug: in-freundlich-adsorption-isotherm-the-value-of-1-n-is-57743
tags: ['Chemistry', 'Surface Chemistry', 'BITSAT', 'BITSAT 2019']
source_db: MarksBatch2_P1.db
question:
<div class="question">In Freundlich adsorption isotherm, the value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn><mo>/</mo><mi mathvariant="normal">n</mi></math> is</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">between <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn></math> in all cases</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">between <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>4</mn></math> in all cases</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">always <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn></math> in case of physical adsorption</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">always <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn></math> in use of chemical adsorption</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">between <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn></math> in all cases</span> </div>

answer:
<div class="solution"><p>Freundlich isotherm is related with the pressure of gas adsorbed, as follows <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mi mathvariant="normal">x</mi><mi mathvariant="normal">m</mi></mfrac><mo>=</mo><mi mathvariant="normal">k</mi><mo>·</mo><msup><mi mathvariant="normal">p</mi><mrow><mn>1</mn><mo>/</mo><mi mathvariant="normal">n</mi></mrow></msup></math></p><p>where, <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mi mathvariant="normal">x</mi><mi mathvariant="normal">m</mi></mfrac><mo>=</mo></math> amount of gas <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mi mathvariant="normal">x</mi></mfenced></math> adsorbed over the surface of mass <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mi mathvariant="normal">m</mi></mfenced></math>.</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">p</mi><mo>=</mo></math> pressure of gas</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">k</mi><mo>=</mo></math> freundlich constant</p><p>Thus, value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>'</mo><mi mathvariant="normal">n</mi><mo>'</mo></math> is always greater than 1 and therefore value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn><mo>/</mo><mi mathvariant="normal">n</mi></math> lies between <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn></math>.</p></div>


tags: ['Mathematics', 'Permutation Combination', 'BITSAT', 'BITSAT 2013']
source_db: MarksBatch2_P1.db
title:
in-how-many-ways-can-5-boys-and-5-girls-be-seated-at-a-round-table-so-that-no-two-girls-may-be-together

slug:
in-how-many-ways-can-5-boys-and-5-girls-be-seated-at-a-round-table-so-that-no-two-girls-may-be-together-67260

question:
<div class="question">In how many ways can 5 boys and 5 girls be seated at a round table so that no two girls may be together?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$4 !$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$5 !$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$4 !+5 !$</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">$4 ! \times 5 !$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$4 ! \times 5 !$</span> </div>

answer:
<div class="solution">Leaving one seat vacant between two boys, 5 boys may be seated in $4 !$ ways. Then at remaining 5 seats, 5 girls any sit in $5 !$ ways. Hence the required number $=4 ! \times 5 !$</div>


tags: ['Mathematics', 'Permutation Combination', 'BITSAT', 'BITSAT 2012']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">1024</span> </div>
source_db: MarksBatch2_P1.db
title:
in-how-many-ways-can-5-prizes-be-distributed-among-4-boys-when-every-boy-can-take-one-or-more-prizes

slug:
in-how-many-ways-can-5-prizes-be-distributed-among-4-boys-when-every-boy-can-take-one-or-more-prizes-17147

question:
<div class="question">In how many ways can 5 prizes be distributed among 4 boys when every boy can take one or more prizes?</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">1024</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">625</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">120</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">600</span> </li> </ul>

answer:
<div class="solution">First prize may be given to any one of the 4 boys, hence first prize can be distributed in 4 ways. similarly every one of second, third, fourth and fifth prizes can also be given in 4 ways. the number of ways of their distribution $=4 \times 4 \times 4 \times 4 \times 4=4^{5}=1024$</div>


title: in-how-many-ways-can-a-committee-of-5-made-out-6-men-and-4-women-containing-atleast-one-woman
slug: in-how-many-ways-can-a-committee-of-5-made-out-6-men-and-4-women-containing-atleast-one-woman-83524
tags: ['Mathematics', 'Permutation Combination', 'BITSAT', 'BITSAT 2014']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">246</span> </div>
source_db: MarksBatch2_P1.db
question:
<div class="question">In how many ways can a committee of 5 made out 6 men and 4 women containing atleast one woman?</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">246</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">222</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">186</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">None of these</span> </li> </ul>

answer:
<div class="solution">A committee of 5 out of $6+4=10$ can be made in ${ }^{10} \mathrm{C}_{5}=252$ ways. If no woman is to be included, thennumber of ways $={ }^{5} \mathrm{C}_{5}=6$
<br/> <br/>$\therefore$ the required number $=252-6=246$</div>


tags: ['Chemistry', 'Biomolecules', 'BITSAT', 'BITSAT 2018']
source_db: MarksBatch2_P1.db
title:
in-nonreducing-disaccharide-the-reducing-group-of-monosaccharides-ie-aldehydic-or-ketonic-group-are-bounded-which-of-the-following-disaccharide-is-a-n

slug:
in-nonreducing-disaccharide-the-reducing-group-of-monosaccharides-ie-aldehydic-or-ketonic-group-are-bounded-which-of-the-following-disaccharide-is-a-n-24922

question:
<div class="question"><p>In non-reducing disaccharide, the reducing group of monosaccharides i.e. aldehydic or ketonic group are bounded.</p><p>Which of the following disaccharide is a non-reducing sugar?</p></div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><img src="https://cdn-question-pool.getmarks.app/pyq/bitsat/ad4ba0e6-44fc-4e5f-ac13-1d28498a5a1a-image.png" style="width: 428px; height: 169px;"/></span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data"><img src="https://cdn-question-pool.getmarks.app/pyq/bitsat/9d5e6421-d71b-4d6a-9618-107da7a7da96-image.png" style="width: 414px; height: 169px;"/></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data"><img src="https://cdn-question-pool.getmarks.app/pyq/bitsat/3a2ff1e6-c8ac-4133-81b3-889ce8bb7698-image.png" style="width: 447px; height: 172px;"/></span> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><img src="https://cdn-question-pool.getmarks.app/pyq/bitsat/28aa40f7-61cb-45e2-b795-2e7a42d40116-image.png" style="width: 460px; height: 172px;"/></span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><img src="https://cdn-question-pool.getmarks.app/pyq/bitsat/9d5e6421-d71b-4d6a-9618-107da7a7da96-image.png" style="width: 414px; height: 169px;"/></span> </div>

answer:
<div class="solution"><p><img src="https://cdn-question-pool.getmarks.app/pyq/bitsat/2dcb5ff4-85e4-492f-b1e4-31a14c01f623-image.png" style="width: 446px; height: 199px;"/></p><p>This structure represents sucrose in which <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">α</mi><mo>-</mo><mi mathvariant="normal">D</mi><mo>-</mo></math>glucose and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>β</mi><mo>-</mo><mi mathvariant="normal">D</mi><mo>-</mo></math>fructose is attached to each other by <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi mathvariant="normal">C</mi><mn>1</mn></msub><mo>-</mo><msub><mi mathvariant="normal">C</mi><mn>2</mn></msub></math> glycosidic linkage. Since reducing groups of glucose and fructose are involved in glycosidic bond formation, this is considered as non-reducing sugar.</p></div>


title: in-paper-chromatography
slug: in-paper-chromatography-14282
question: <div class="question">In paper chromatography:</div>
tags: ['Chemistry', 'General Organic Chemistry', 'BITSAT', 'BITSAT 2012']
source_db: MarksBatch2_P1.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">Mobile phase is liquid and stationary phase is solid.</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">Mobile phase is solid and stationary phase is liquid.</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">Both phases are liquids.</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">Both phases are solids.</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">Both phases are liquids.</span> </div>

answer:
<div class="solution">Paper chromatography is a special case of partition chromatography where the special quality paper containing water trapped in it acts as a stationary phase and solvent as a mobile phase. Thus, both phases are liquids.</div>


title: in-po-4-3-the-formal-charge-on-each-oxygen-atom-and-the-p-o-bond-order-respectively-are-1
slug: in-po-4-3-the-formal-charge-on-each-oxygen-atom-and-the-p-o-bond-order-respectively-are-1-43688
tags: ['Chemistry', 'Chemical Bonding and Molecular Structure', 'JEE Main']
source_db: MarksBatch2_P1.db
question:
<div class="question">In $\mathrm{PO}_{4}^{3-},$ the formal charge on each oxygen atom and the $\mathrm{P}-\mathrm{O}$ bond order respectively are</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">-0.75,0.6</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">-0.75,1.0</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">-0.75,1.25</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">-3,1.25</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">-0.75,1.25</span> </div>

answer:
<div class="solution"><img src="https://cdn-question-pool.getmarks.app/pyq/bitsat/tA7unZEkDn7bmgq12DkXbxTD4nGXISHeFNZ86LWXnlY.original.fullsize.png"/><br/> <br/> <br/>Bond order
<br/> <br/>$=\frac{\text { Number of bonds }}{\text { Number of Resonating structures }}$
<br/> <br/>$=\frac{5}{4}=1.25$
<br/> <br/>Three unit negative charge is being shared by four $\mathrm{O}$ atoms. Formal charge $=-3 / 4$</div>


title: in-reaction-bf-3-3-libh-4-3-lif-x-x-is
slug: in-reaction-bf-3-3-libh-4-3-lif-x-x-is-49975
tags: ['Chemistry', 'p Block Elements (Group 13 & 14)', 'BITSAT', 'BITSAT 2023 (Memory Based Paper 2)']
source_db: MarksBatch2_P1.db
question:
<div class="question">In reaction $\mathrm{BF}_3+3 \mathrm{LiBH}_4 \rightarrow 3 \mathrm{LiF}+X ; X$ is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\mathrm{B}_4 \mathrm{H}_{10}$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$\mathrm{B}_2 \mathrm{H}_6$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\mathrm{BH}_3$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\mathrm{B}_3 \mathrm{H}_8$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\mathrm{B}_2 \mathrm{H}_6$</span> </div>

answer:
<div class="solution">$2 \mathrm{BF}_3+3 \mathrm{LiBH}_4 \rightarrow 2 \mathrm{~B}_2 \mathrm{H}_6+3 \mathrm{LiF}$</div>


title: in-redox-reaction-1-g-eq-of-reducing-agent-requires-pgm-eq-of-oxidising-agent-the-value-of-p-is
tags: ['Chemistry', 'Redox Reactions', 'BITSAT', 'BITSAT 2010']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">1</span> </div>
source_db: MarksBatch2_P1.db
slug:
in-redox-reaction-1-g-eq-of-reducing-agent-requires-pgm-eq-of-oxidising-agent-the-value-of-p-is-83534

question:
<div class="question">In redox reaction $1 \mathrm{~g}$ -eq of reducing agent requires $\mathrm{Pgm}-\mathrm{eq} .$ of oxidising agent. The value of $\mathrm{P}$ is</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">1</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">2</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">3</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">Depends on reaction</span> </li> </ul>

answer:
<div class="solution">In redox reaction, $g$ equivalent of reducing agent $=\mathrm{g}$. equivalent of oxidising agent Hence lg equ. of reducing agent $=\mathrm{Pg}$ equ. of oxidising agent.</div>


tags: ['Physics', 'Alternating Current', 'BITSAT', 'BITSAT 2020']
answer: <div class="solution">No Solution</div>
source_db: MarksBatch2_P1.db
title:
in-series-combination-of-r-l-and-c-with-an-ac-source-at-resonance-if-r-20-ohm-then-impedence-z-of-the-combination-is

slug:
in-series-combination-of-r-l-and-c-with-an-ac-source-at-resonance-if-r-20-ohm-then-impedence-z-of-the-combination-is-46744

question:
<div class="question">In series combination of $R, L$ and $C$ with an A.C. source at resonance, if $\mathrm{R}=20 \mathrm{ohm},$ then impedence $Z$ of the combination is</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$20 \mathrm{ohm}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">zero</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">10 ohm</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$400 \mathrm{ohm}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$20 \mathrm{ohm}$</span> </div>


tags: ['Chemistry', 'General Organic Chemistry', 'BITSAT', 'BITSAT 2011']
source_db: MarksBatch2_P1.db
title:
in-the-anion-hcoo-the-two-carbonoxygen-bonds-are-found-to-be-of-equal-length-what-is-the-reason-for-it-1

slug:
in-the-anion-hcoo-the-two-carbonoxygen-bonds-are-found-to-be-of-equal-length-what-is-the-reason-for-it-1-36805

question:
<div class="question">In the anion $\mathrm{HCOO}^{-}$ the two carbon-oxygen bonds are found to be of equal length. What is the reason for it?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">The $\mathrm{C}=\mathrm{O}$ bond is weaker than the $\mathrm{C}-\mathrm{O}$ bond.</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">The anion $\mathrm{HCOO}^{-}$ has two resonating structures.</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">The anion is obtained byremoval of a proton from the acid molecule.</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">Electronic orbitals of carbon atom are hybridised.</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">The anion $\mathrm{HCOO}^{-}$ has two resonating structures.</span> </div>

answer:
<div class="solution"><img src="https://cdn-question-pool.getmarks.app/pyq/bitsat/ccZH-QVFFp4nOA6a0li2mleynlx7S7UfKm5sWcgQafc.original.fullsize.png"/></div>


tags: ['Physics', 'Current Electricity', 'BITSAT', 'BITSAT 2022']
source_db: MarksBatch2_P1.db
title:
in-the-arrangement-shown-in-figure-when-the-switch-s-2-is-open-the-galvanometer-shows-no-deflection-for-l-50-cm-when-the-switch-s-2-is-closed-the-galv

slug:
in-the-arrangement-shown-in-figure-when-the-switch-s-2-is-open-the-galvanometer-shows-no-deflection-for-l-50-cm-when-the-switch-s-2-is-closed-the-galv-74187

question:
<div class="question">In the arrangement shown in figure, when the switch $S_2$ is open, the galvanometer, shows no deflection for $l=50 \mathrm{~cm}$ when the switch $S_2$ is closed, the galvanometer shows no deflection for $l=0.416 \mathrm{~m}$. The internal resistance $(r)$ of $6 \mathrm{~V}$ cell is
<br/><img src="https://cdn-question-pool.getmarks.app/pyq/bitsat/RMx7mHemGAXNHHKGr7KSr50nRFhQiPlDYK2aY2iwQck.original.fullsize.png"/><br/></div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data"> <br/>$2 \Omega$
<br/></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data"> <br/>$3 \Omega$
<br/></span> </li><li class=""> <span class="option-label">C</span> <span class="option-data"> <br/>$5 \Omega$
<br/></span> </li><li class=""> <span class="option-label">D</span> <span class="option-data"> <br/>$9 \Omega$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"> <br/>$2 \Omega$
<br/></span> </div>

answer:
<div class="solution">When $S_2$ open.
<br/>Assume resistance of $X Y=R$.
<br/>Resistance of wire per unit length, $x=\frac{R}{L}=R \Omega \mathrm{m}^{-1}$
<br/>$\because \quad I=E_0 / R$
<br/>Now, the potential drop across $50 \mathrm{~cm}$ length is $6 \mathrm{~V}$, So
<br/>$$
<br/>\begin{array}{cc} 
<br/>&amp; \frac{E_0}{R} \times R \times \frac{50}{100}=6 \\
<br/>\Rightarrow &amp; E_0=12 \mathrm{~V}
<br/>\end{array}
<br/>$$
<br/>When $S_2$ closed, potential drop across $0.416 \mathrm{~cm}$ length,
<br/>$$
<br/>V_1=\frac{E_0}{R} \times R \times 0.416=12 \times 0.416 \approx 5 \mathrm{~V}
<br/>$$
<br/>Hence, $\quad E-I r=5 \mathrm{~V}$
<br/>$$
<br/>\begin{aligned}
<br/>\Rightarrow &amp; &amp; 6-I r &amp; =5 \\
<br/>&amp; \because &amp; I &amp; =\frac{5}{10} \\
<br/>\therefore &amp; &amp; 6-5 &amp; =\frac{5}{10} r \\
<br/>\Rightarrow &amp; &amp; r &amp; =2 \Omega
<br/>\end{aligned}
<br/>$$</div>


tags: ['Mathematics', 'Binomial Theorem', 'BITSAT', 'BITSAT 2011']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">9</span> </div>
source_db: MarksBatch2_P1.db
title:
in-the-binomial-left21-331-3rightmathrmn-if-the-ratio-of-the-seventh-term-from-the-beginning-of-the-expansion-to-the-seventh-term-from-its-end-is-1-6-

slug:
in-the-binomial-left21-331-3rightmathrmn-if-the-ratio-of-the-seventh-term-from-the-beginning-of-the-expansion-to-the-seventh-term-from-its-end-is-1-6-85406

question:
<div class="question">In the binomial \(\left(2^{1 / 3}+3^{-1 / 3}\right)^{\mathrm{n}}\), if the ratio of the seventh term from the beginning of the expansion to the seventh term from its end is \(1 / 6\), then \(n\) equal to</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">6</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">9</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">12</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">15</span> </li> </ul>

answer:
<div class="solution">\(\mathrm{T}_{\mathrm{r}+1}={ }^{\mathrm{n}} \mathrm{C}_{\mathrm{r}} a^{\mathrm{n}-\mathrm{r}} \cdot \mathrm{b}^{\mathrm{r}}\) where \(\mathrm{a}=2^{1 / 3}\) and \(\mathrm{b}=3^{-1 / 3}\)<br/>\(\mathrm{T}_7\) from beginning \(={ }^n \mathrm{C}_6 a^{\mathrm{n}-6} \mathrm{~b}^6\) and \(\mathrm{T}_7\) from end \(={ }^{\mathrm{n}} \mathrm{C}_6 \quad \mathrm{~b}^{\mathrm{n}-6} \quad \mathrm{a}^6\)<br/>\(\begin{aligned}<br/>&amp; \Rightarrow \frac{\mathrm{a}^{\mathrm{n}-12}}{\mathrm{~b}^{\mathrm{n}-12}}=\frac{1}{6} \Rightarrow 2^{\frac{\mathrm{n}-12}{3}} \cdot 3^{\frac{\mathrm{n}-12}{3}}=6^{-1} \\<br/>&amp; \Rightarrow \mathrm{n}-12=-3 \Rightarrow \mathrm{n}=9<br/>\end{aligned}\)</div>


tags: ['Chemistry', 'Electrochemistry', 'BITSAT', 'BITSAT 2014']
answer: <div class="solution">No Solution</div>
source_db: MarksBatch2_P1.db
title:
in-the-cell-reaction-cu-s-2-ag-aq-cu-2-aq-2-ag-s-e-cell-0-046-v-by-doubling-the-concentration-of-cu-2-e-cell-0-is

slug:
in-the-cell-reaction-cu-s-2-ag-aq-cu-2-aq-2-ag-s-e-cell-0-046-v-by-doubling-the-concentration-of-cu-2-e-cell-0-is-96244

question:
<div class="question">In the cell reaction
<br/> <br/>$\mathrm{Cu}(\mathrm{s})+2 \mathrm{Ag}^{+}(\mathrm{aq}) \longrightarrow \mathrm{Cu}^{2+}(\mathrm{aq})+2 \mathrm{Ag}(\mathrm{s})$
<br/> <br/>$\mathrm{E}_{\text {cell }}^{0}=0.46 \mathrm{~V}$. By doubling the concentration of $\mathrm{Cu}^{2+}, \mathrm{E}_{\text {cell }}^{0}$ is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">doubled</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">halved</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">increases but less than double</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">decreases by a small fraction</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">decreases by a small fraction</span> </div>


title: in-the-circuit-given-below-the-charge-in-c-on-the-capacitor-having-5-f-is-1
slug: in-the-circuit-given-below-the-charge-in-c-on-the-capacitor-having-5-f-is-1-96371
tags: ['Physics', 'Capacitance', 'JEE Main']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">9</span> </div>
source_db: MarksBatch2_P1.db
question:
<div class="question">In the circuit given below, the charge in $\mu \mathrm{C},$ on the capacitor having $5 \mu \mathrm{F}$ is
<br/> <br/><img src="https://cdn-question-pool.getmarks.app/pyq/bitsat/O33zcfEW8hFj-ynlLZAV1V0aMC-qTMsxK-IC-6pbWLM.original.fullsize.png"/></div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">4.5</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">9</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">7</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">15</span> </li> </ul>

answer:
<div class="solution">Potential difference across the branch de is
<br/> <br/>$6 \mathrm{~V}$. Net capacitance of de branch is $2.1 \mu \mathrm{F}$ So, $q=C V$ $\Rightarrow \mathrm{q}=2.1 \times 6 \mu \mathrm{C} \quad \Rightarrow \mathrm{q}=12.6 \mu \mathrm{C}$
<br/> <br/>Potential across $3 \mu \mathrm{F}$ capacitance is $\mathrm{V}=\frac{12.6}{3}=4.2 \mathrm{volt}$
<br/> <br/>Potential across 2 and 5 combination in parallel is $6-4.2=1.8 \mathrm{~V}$
<br/> <br/>So, $q^{\prime}=(1.8)(5)=9 \mu C$</div>


tags: ['Physics', 'Current Electricity', 'BITSAT', 'BITSAT 2018']
source_db: MarksBatch2_P1.db
title:
in-the-circuit-given-below-the-value-of-resistance-x-when-the-potential-difference-between-the-points-b-and-d-is-zero-will-be

slug:
in-the-circuit-given-below-the-value-of-resistance-x-when-the-potential-difference-between-the-points-b-and-d-is-zero-will-be-33347

question:
<div class="question">In the circuit given below, the value of resistance <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>X</mi></math>, when the potential difference between the points <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>B</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>D</mi></math> is zero, will be<br/><img src="https://cdn-question-pool.getmarks.app/pyq/bitsat/62e85709-d735-4db3-bf90-b54978008a7a-194c46af-8a17-46a9-a151-76d06f2d8091-image.png" style="width: 300px; height: 291px;"/></div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>9</mn><mo> </mo><mi mathvariant="normal">Ω</mi></math></span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>8</mn><mo> </mo><mi mathvariant="normal">Ω</mi></math></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>4</mn><mo> </mo><mi mathvariant="normal">Ω</mi></math></span> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>6</mn><mo> </mo><mi mathvariant="normal">Ω</mi></math></span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>8</mn><mo> </mo><mi mathvariant="normal">Ω</mi></math></span> </div>

answer:
<div class="solution"><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi><mo>=</mo><mn>15</mn><mo>+</mo><mn>6</mn><mo>=</mo><mn>21</mn><mo> </mo><mi mathvariant="normal">Ω</mi><mo>,</mo><mo> </mo><mi>Q</mi><mo>=</mo><mfrac><mrow><mn>8</mn><mi>X</mi></mrow><mrow><mn>8</mn><mo>+</mo><mi>X</mi></mrow></mfrac><mo>+</mo><mn>3</mn></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>R</mi><mo>=</mo><mn>15</mn><mo>+</mo><mfrac><mrow><mn>6</mn><mo>×</mo><mn>6</mn></mrow><mrow><mn>6</mn><mo>+</mo><mn>6</mn></mrow></mfrac><mo>=</mo><mn>18</mn><mo> </mo><mi mathvariant="normal">Ω</mi><mo>,</mo><mo> </mo><mi>S</mi><mo>=</mo><mn>4</mn><mo>+</mo><mfrac><mrow><mn>4</mn><mo>×</mo><mn>4</mn></mrow><mrow><mn>4</mn><mo>+</mo><mn>4</mn></mrow></mfrac><mo>=</mo><mn>6</mn><mo> </mo><mi mathvariant="normal">Ω</mi></math></p><p>Since potential difference across <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>B</mi><mi>D</mi></math> is zero hence it satisfies Wheatstone's Bridge condition,</p><p> <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>∴</mo><mfrac><mi>P</mi><mi>Q</mi></mfrac><mo>=</mo><mfrac><mi>R</mi><mi>S</mi></mfrac><mo>,</mo></math> so <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>Q</mi><mo>=</mo><mfrac><mrow><mi>P</mi><mi>S</mi></mrow><mi>R</mi></mfrac><mo>=</mo><mfrac><mrow><mn>21</mn><mo>×</mo><mn>6</mn></mrow><mn>18</mn></mfrac><mo>=</mo><mn>7</mn><mo> </mo><mi mathvariant="normal">Ω</mi></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>3</mn><mo>+</mo><mfrac><mrow><mn>8</mn><mi>x</mi></mrow><mrow><mn>8</mn><mo>+</mo><mi>x</mi></mrow></mfrac><mo>=</mo><mn>7</mn><mo>⇒</mo><mi>x</mi><mo>=</mo><mn>8</mn><mo> </mo><mi mathvariant="normal">Ω</mi></math></p></div>


tags: ['Physics', 'Semiconductors', 'BITSAT', 'BITSAT 2022']
source_db: MarksBatch2_P1.db
title:
in-the-circuit-shown-assume-the-diode-to-be-ideal-when-v-i-increases-from-2-v-to-6-v-the-change-in-current-is-in-ma

slug:
in-the-circuit-shown-assume-the-diode-to-be-ideal-when-v-i-increases-from-2-v-to-6-v-the-change-in-current-is-in-ma-46051

question:
<div class="question">In the circuit shown assume the diode to be ideal. When $V_i$ increases from $-2 \mathrm{~V}$ to $6 \mathrm{~V}$, the change in current is (in $\mathrm{mA})$
<br/><img src="https://cdn-question-pool.getmarks.app/pyq/bitsat/BRTAX7F05BNX2CwmhPaBc38waP7UU-NpysSepexmcWg.original.fullsize.png"/><br/></div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"> <br/>Zero 
<br/></span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data"> <br/>$20$
<br/></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data"> <br/>$\frac{25}{8}$
<br/></span> </li><li class=""> <span class="option-label">D</span> <span class="option-data"> <br/>$32$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"> <br/>$20$
<br/></span> </div>

answer:
<div class="solution">$$
<br/>l_{\text {initial }}=0 \text { for } V_i \leq+1 \mathrm{~V}
<br/>$$
<br/>This diode conduct only beyond $V_i=+1 \mathrm{~V}$
<br/>$$
<br/>I_{\text {final }}=\frac{5}{250}=0.02 \mathrm{~A}
<br/>$$
<br/>So, change in $I=0.02 \mathrm{~A}=20 \mathrm{~mA}$</div>


tags: ['Physics', 'Alternating Current', 'JEE Main']
source_db: MarksBatch2_P1.db
title:
in-the-circuit-shown-below-the-ac-source-has-voltage-v-20-cos-t-volt-with-2000-rad-s-the-amplitude-of-the-current-will-be-nearest-to-1

slug:
in-the-circuit-shown-below-the-ac-source-has-voltage-v-20-cos-t-volt-with-2000-rad-s-the-amplitude-of-the-current-will-be-nearest-to-1-62728

question:
<div class="question">In the circuit shown below, the ac source has voltage $V=20 \cos (\omega t)$ volt with $\omega=2000 \mathrm{rad} / \mathrm{s}$
<br/> <br/>The amplitude of the current will be nearest to
<br/> <br/><img src="https://cdn-question-pool.getmarks.app/pyq/bitsat/0sMCMm8Dj6GVncaD-q5TZfNMH0aQT3lyO35PZhANVqc.original.fullsize.png"/></div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$2 \mathrm{~A}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$3.3 \mathrm{~A}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$2 / \sqrt{5} \mathrm{~A}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\sqrt{5} \mathrm{~A}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$2 \mathrm{~A}$</span> </div>

answer:
<div class="solution">$$
<br/> <br/>\begin{aligned}
<br/> <br/>Z=&amp; \sqrt{R^{2}+\left(\omega L-\frac{1}{\omega C}\right)^{2}} \\
<br/> <br/>&amp;=\sqrt{10^{2}+\left(2000 \times 5 \times 10^{3}-\frac{1}{2000 \times 50 \times 10^{-6}}\right)^{2}} \\
<br/> <br/>=&amp; 10 \Omega \\
<br/> <br/>\quad i=\frac{V_{0}}{Z}=\frac{20}{10}=2 A .
<br/> <br/>\end{aligned}
<br/> <br/>$$</div>


tags: ['Physics', 'Alternating Current', 'BITSAT', 'BITSAT 2015']
source_db: MarksBatch2_P1.db
title:
in-the-circuit-shown-below-the-ac-source-has-voltage-v-20-cos-t-volt-with-2000-rad-s-the-amplitude-of-the-current-will-be-nearest-to

slug:
in-the-circuit-shown-below-the-ac-source-has-voltage-v-20-cos-t-volt-with-2000-rad-s-the-amplitude-of-the-current-will-be-nearest-to-12923

question:
<div class="question">In the circuit shown below, the ac source has voltage $V=20 \cos (\omega t)$ volt with $\omega=2000 \mathrm{rad} / \mathrm{s}$
<br/> <br/>The amplitude of the current will be nearest to
<br/> <br/><img src="https://cdn-question-pool.getmarks.app/pyq/bitsat/0sMCMm8Dj6GVncaD-q5TZfNMH0aQT3lyO35PZhANVqc.original.fullsize.png"/></div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$2 \mathrm{~A}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$3.3 \mathrm{~A}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$2 / \sqrt{5} \mathrm{~A}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\sqrt{5} \mathrm{~A}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$2 \mathrm{~A}$</span> </div>

answer:
<div class="solution">$$
<br/> <br/>\begin{aligned}
<br/> <br/>Z=&amp; \sqrt{R^{2}+\left(\omega L-\frac{1}{\omega C}\right)^{2}} \\
<br/> <br/>&amp;=\sqrt{10^{2}+\left(2000 \times 5 \times 10^{3}-\frac{1}{2000 \times 50 \times 10^{-6}}\right)^{2}} \\
<br/> <br/>=&amp; 10 \Omega \\
<br/> <br/>\quad i=\frac{V_{0}}{Z}=\frac{20}{10}=2 A .
<br/> <br/>\end{aligned}
<br/> <br/>$$</div>


tags: ['Physics', 'Current Electricity', 'JEE Main']
source_db: MarksBatch2_P1.db
title:
in-the-circuit-shown-in-figure-potential-difference-between-points-a-and-b-is-16-v-the-current-passing-through-2-resistance-will-be-1

slug:
in-the-circuit-shown-in-figure-potential-difference-between-points-a-and-b-is-16-v-the-current-passing-through-2-resistance-will-be-1-30166

question:
<div class="question">In the circuit shown in figure potential difference between points $A$ and $B$ is $16 \mathrm{~V}$. the current passing through $2 \Omega$ resistance will be
<br/> <br/><img src="https://cdn-question-pool.getmarks.app/pyq/bitsat/UZbUESIx80GM6K0l50vBIuvRhaCpq4e_7uhL3_sImFs.original.fullsize.png"/></div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$2.5 \mathrm{~A}$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$3.5 \mathrm{~A}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$4.0 \mathrm{~A}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">zero</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$3.5 \mathrm{~A}$</span> </div>

answer:
<div class="solution">$\therefore 4 \mathrm{i}_{1}+2\left(\mathrm{i}_{1}+\mathrm{i}_{2}\right)-3+4 \mathrm{i}_{1}=16 \mathrm{~V}$
<br/> <br/>Using Kirchhoff's second law in the closed loop we have
<br/> <br/>$$
<br/> <br/>9-i_{2}-2\left(i_{1}+i_{2}\right)=0
<br/> <br/>$$
<br/> <br/>Solving equations (i) and (ii), we get $\mathrm{i}_{1}=1.5 \mathrm{~A}$ and $\mathrm{i}_{2}=2 \mathrm{~A}$
<br/> <br/>$\therefore$ current through $2 \mathrm{~W}$ resistor $=2+1.5=3.5 \mathrm{~A}$</div>


